{"id":55080,"date":"2025-01-10T22:14:46","date_gmt":"2025-01-10T14:14:46","guid":{"rendered":"https:\/\/swarma.org\/?p=55080"},"modified":"2025-01-10T22:14:46","modified_gmt":"2025-01-10T14:14:46","slug":"%e5%a5%87%e5%bc%82%e5%80%bc%e5%88%86%e8%a7%a3%ef%bc%88svd%ef%bc%89%ef%bd%9c%e9%9b%86%e6%99%ba%e7%99%be%e7%a7%91","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=55080","title":{"rendered":"\u5947\u5f02\u503c\u5206\u89e3\uff08SVD\uff09\uff5c\u96c6\u667a\u767e\u79d1"},"content":{"rendered":"
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\u5728\u7ebf\u6027\u4ee3\u6570\u4e2d\uff0c\u5947\u5f02\u503c\u5206\u89e3\uff08Singular value decomposition\uff09\u662f\u4e00\u79cd\u901a\u8fc7\u65cb\u8f6c\u3001\u7f29\u653e\u548c\u518d\u6b21\u65cb\u8f6c\u6765\u5bf9\u5b9e\u77e9\u9635\uff08real matrix \u5143\u7d20\u90fd\u662f\u5b9e\u6570\u7684\u77e9\u9635\uff09\u6216\u590d\u77e9\u9635\uff08complex matrix \u5143\u7d20\u6709\u590d\u6570\u7684\u77e9\u9635\uff09\u8fdb\u884c\u56e0\u5f0f\u5206\u89e3\u7684\u65b9\u6cd5\uff0c\u5982\u4e0b\u56fe\u3002\u5b83\u628a\u5177\u6709\u6807\u51c6\u6b63\u4ea4\u57fa\uff08orthonormal eigenbasis\uff09\u7684<\/strong><\/span>\u65b9\u9635<\/strong>\u7279\u5f81\u5206\u89e3<\/strong>\uff08eigen decomposition \u77e9\u9635\u5206\u89e3\u4e3a<\/strong><\/span>\u7279\u5f81\u5411\u91cf<\/strong>\u548c<\/strong><\/span>\u7279\u5f81\u503c<\/strong>\uff09\u63a8\u5e7f\u5230\u4e86\u4efb\u610f <\/strong><\/span>m<\/strong>\u00d7<\/strong>n<\/strong>\u77e9\u9635\uff0c\u5e76\u4e0e<\/strong><\/span>\u6781\u5206\u89e3<\/strong>\uff08polar decomposition\uff09\u5bc6\u5207\u76f8\u5173\u3002<\/strong><\/span><\/span><\/p>\n


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\u5177\u4f53\u800c\u8a00\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4e00\u4e2a <\/strong>m<\/strong>\u00d7<\/strong>n<\/strong> \u590d\u77e9\u9635 <\/strong>M <\/strong>\u5206\u89e3\u4e3a <\/strong>M<\/strong>=<\/strong>U<\/strong>\u03a3<\/strong>V<\/strong>\u2217<\/strong><\/sup>\u3002\u5728\u8fd9\u91cc\uff0c<\/strong>U<\/strong>\u662f <\/strong>m<\/strong>\u00d7<\/strong>m<\/strong> \u590d\u9149\u77e9\u9635\uff08complex unitary matrix \u5171\u8f6d\u8f6c\u7f6e\u4e3a\u5176\u9006\u7684\u590d\u6570\u77e9\u9635\uff09\uff0c<\/strong>\u03a3<\/strong> \u662f <\/strong>m<\/strong>\u00d7<\/strong>n<\/strong>\u77e9\u5f62\u5bf9\u89d2\u77e9\u9635\uff08rectangular diagonal matrix\uff09\uff0c\u5176\u5bf9\u89d2\u7ebf\u5143\u7d20\u4e3a\u975e\u8d1f\u5b9e\u6570\uff0c<\/strong>V<\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f <\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>n<\/strong><\/span><\/mi>\u00d7<\/strong><\/span><\/mo>n<\/strong><\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u590d\u9149\u77e9\u9635\uff0c\u800c <\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/strong><\/span><\/mi><\/mrow>\u2217<\/strong><\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f <\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5171\u8f6d\u8f6c\u7f6e\uff08conjugate transpose\uff09\u3002\u8fd9\u79cd\u5206\u89e3\u9002\u7528\u4e8e\u4efb\u4f55\u590d\u77e9\u9635\u3002\u82e5 <\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e3a\u5b9e\u77e9\u9635\uff0c\u5219 <\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5fc5\u4e3a\u5b9e\u6b63\u4ea4\u77e9\u9635\uff08real orthogonal matrices\uff09\uff1b\u6b64\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u5c06SVD\u8868\u793a\u4e3a <\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/strong><\/span><\/mi><\/mrow>=<\/strong><\/span><\/mo>U<\/strong><\/span><\/mi>\u03a3<\/strong><\/span><\/mi>V<\/strong><\/span><\/mi><\/mrow>T<\/strong><\/span><\/sup><\/mi><\/mrow><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002<\/strong><\/span><\/section>\n

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<\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5bf9\u89d2\u5143\u7d20 <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/strong><\/span><\/mi>i<\/span><\/strong><\/span><\/sub><\/mi><\/msub>=<\/span><\/strong><\/span><\/mo>\u03a3<\/span><\/strong><\/span><\/mi>i<\/span><\/strong><\/span><\/sub><\/mi>i<\/span><\/strong><\/span><\/sub><\/mi><\/mrow><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7531 <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u552f\u4e00\u786e\u5b9a\uff0c\u79f0\u4e3a <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5947\u5f02\u503c\u3002\u975e\u96f6\u5947\u5f02\u503c\u7684\u6570\u91cf\u7b49\u4e8e <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684<\/span><\/strong>\u79e9<\/strong>\uff08rank\uff09\u3002\u6211\u4eec\u628a <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5217\u548c <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5217\u5206\u522b\u53eb\u505a <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5de6\u5947\u5f02\u5411\u91cf\u548c\u53f3\u5947\u5f02\u5411\u91cf\u3002\u5b83\u4eec\u5206\u522b\u6784\u6210\u4e24\u7ec4<\/span><\/strong>\u6807\u51c6\u6b63\u4ea4\u57fa<\/strong>\uff08orthonormal bases\uff09 <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>u<\/span><\/strong><\/span><\/mi><\/mrow>1<\/span><\/strong><\/span><\/sub><\/mn><\/msub>,<\/span><\/strong><\/span><\/mo>\u2026<\/span><\/strong><\/span><\/mo>,<\/span><\/strong><\/span><\/mo>u<\/span><\/strong><\/span><\/mi><\/mrow>m<\/span><\/strong><\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>v<\/span><\/strong><\/span><\/mi><\/mrow>1<\/span><\/strong><\/span><\/sub><\/mn><\/msub>,<\/span><\/strong><\/span><\/mo>\u2026<\/span><\/strong><\/span><\/mo>,<\/span><\/strong><\/span><\/mo>v<\/span><\/strong><\/span><\/mi><\/mrow>n<\/span><\/strong><\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u5982\u679c\u6211\u4eec\u5c06\u503c\u4e3a\u96f6\u7684\u5947\u5f02\u503c <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/strong><\/span><\/mi>i<\/span><\/strong><\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> <\/span><\/strong><\/span><\/sub>\u6392\u5728\u6700\u540e\uff0c\u5947\u5f02\u503c\u5206\u89e3\u5c31\u53ef\u4ee5\u5199\u6210\uff1a<\/span><\/strong><\/span><\/section>\n

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\u8fd9\u91cc <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>r<\/span><\/strong><\/span><\/mi>\u2264<\/span><\/strong><\/span><\/mo>min<\/span><\/strong><\/span><\/mo>{<\/span><\/strong><\/span><\/mo>m<\/span><\/strong><\/span><\/mi>,<\/span><\/strong><\/span><\/mo>n<\/span><\/strong><\/span><\/mi>}<\/span><\/strong><\/span><\/mo><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u79e9\u3002<\/span><\/strong><\/span><\/section>\n

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\u867d\u7136SVD\u4e0d\u662f\u552f\u4e00\u7684\uff0c\u4f46\u6211\u4eec\u603b\u662f\u53ef\u4ee5\u9009\u62e9\u8ba9\u5947\u5f02\u503c <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/strong><\/span><\/mi>i<\/span><\/strong><\/span><\/mi>i<\/span><\/strong><\/span><\/mi><\/mrow><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u6309\u964d\u5e8f\u6392\u5217\u3002\u8fd9\u6837\u4e00\u6765\uff0c<\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff08\u800c\u975e <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff09\u5c31\u7531 <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u552f\u4e00\u786e\u5b9a\u4e86\u3002<\/span><\/strong><\/span><\/section>\n

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\u6709\u65f6\uff0c\u6211\u4eec\u4e5f\u628aSVD\u79f0\u4e3a\u7d27\u51d1SVD\uff08compact SVD\uff09\u3002\u7d27\u51d1\u578bSVD\u662f\u4e0e\u4e4b\u7c7b\u4f3c\u7684\u4e00\u4e2a\u5206\u89e3\uff0c\u5f62\u5982<\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/strong><\/span><\/mi><\/mrow>=<\/span><\/strong><\/span><\/mo>U<\/span><\/strong><\/span><\/mi>r<\/span><\/strong><\/span><\/sub><\/mi><\/msub>\u03a3<\/span><\/strong><\/span><\/mi>r<\/span><\/strong><\/span><\/sub><\/mi><\/msub>V<\/span><\/strong><\/span><\/mi>r<\/span><\/strong><\/span><\/sub><\/mi><\/msub><\/mrow>\u2217<\/span><\/strong><\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u5176\u4e2d <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>r<\/span><\/strong><\/span><\/mi>\u00d7<\/span><\/strong><\/span><\/mo>r<\/span><\/strong><\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u65b9\u5f62\u5bf9\u89d2\u77e9\u9635\uff0c<\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>r<\/span><\/strong><\/span><\/mi>\u2264<\/span><\/strong><\/span><\/mo>min<\/span><\/strong><\/span><\/mo>{<\/span><\/strong><\/span><\/mo>m<\/span><\/strong><\/span><\/mi>,<\/span><\/strong><\/span><\/mo>n<\/span><\/strong><\/span><\/mi>}<\/span><\/strong><\/span><\/mo><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c<\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>r<\/span><\/strong><\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5c31\u662f <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u79e9\uff0c\u53ea\u5305\u542b\u975e\u96f6\u5947\u5f02\u503c\u3002\u5728\u8fd9\u79cd\u53d8\u4f53\u4e2d\uff0c<\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/strong><\/span><\/mi>\u00d7<\/span><\/strong><\/span><\/mo>r<\/span><\/strong><\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u534a\u9149\u77e9\u9635\uff08semi-unitary matrix \u975e\u65b9\u9635\u7684\u9149\u77e9\u9635\uff09\uff0c<\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>n<\/span><\/strong><\/span><\/mi>\u00d7<\/span><\/strong><\/span><\/mo>r<\/span><\/strong><\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u534a\u9149\u77e9\u9635\uff0c\u6ee1\u8db3 <\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/strong><\/span><\/mi><\/mrow>\u2217<\/span><\/strong><\/span><\/mo><\/msup>U<\/span><\/strong><\/span><\/mi><\/mrow>=<\/span><\/strong><\/span><\/mo>V<\/span><\/strong><\/span><\/mi><\/mrow>\u2217<\/span><\/strong><\/span><\/mo><\/msup>V<\/span><\/strong><\/span><\/mi><\/mrow>=<\/span><\/strong><\/span><\/mo>I<\/span><\/strong><\/span><\/mi><\/mrow>r<\/span><\/strong><\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff08\u5355\u4f4d\u77e9\u9635\uff09\u3002<\/span><\/strong><\/span><\/section>\n

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SVD\u5728\u6570\u5b66\u4e0a\u6709\u591a\u79cd\u5e94\u7528\uff0c\u5305\u62ec\u8ba1\u7b97<\/span><\/strong>\u4f2a\u9006<\/strong>\uff08pseudoinverse\uff09\u3001\u77e9\u9635\u8fd1\u4f3c\uff08matrix approximation\uff09\u4ee5\u53ca\u786e\u5b9a\u77e9\u9635\u7684<\/span><\/strong>\u79e9<\/strong>\uff08rank \u7ebf\u6027\u65e0\u5173\u5411\u91cf\u7684\u6700\u5927\u4e2a\u6570\uff09\u3001<\/span><\/strong>\u503c\u57df<\/strong>\uff08range\uff09\u548c<\/span><\/strong>\u96f6\u7a7a\u95f4<\/strong>\uff08null space\uff09\u3002\u6b64\u5916\uff0cSVD\u5728\u79d1\u5b66\u3001\u5de5\u7a0b\u548c\u7edf\u8ba1\u5b66\u7684\u5404\u4e2a\u9886\u57df\u90fd\u5f88\u6709\u7528\uff0c\u6bd4\u5982<\/span><\/strong>\u4fe1\u53f7\u5904\u7406<\/strong>\uff08signal processing\uff09\u3001<\/span><\/strong>\u6570\u636e\u6700\u5c0f\u4e8c\u4e58\u62df\u5408<\/strong>\uff08least squares fitting of data\uff09\u548c\u8fc7\u7a0b\u63a7\u5236\u7b49\u3002<\/span><\/strong><\/span><\/section>\n

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\u8fd1\u5e74\u6765\uff0c\u5f20\u6c5f\u8001\u5e08\u5e26\u9886\u7814\u7a76\u7ec4\u5f00\u59cb\u805a\u7126\u57fa\u4e8e\u65b0\u5174 AI \u6280\u672f\u8fdb\u884c\u6570\u636e\u9a71\u52a8\u7684\u81ea\u52a8\u5efa\u6a21\u7814\u7a76\uff0c\u5e76\u7acb\u5fd7\u7834\u89e3\u590d\u6742\u7cfb\u7edf\u7684\u6d8c\u73b0\u4e4b\u8c1c\u3002\u6211\u4eec\u5e0c\u671b\u521b\u5efa\u4e00\u4e2a\u53eb\u505a<\/span><\/strong><\/span>\u201c\u590d\u6742AI\u6b21\u65b9\u201d\u7684\u5f00\u653e\u5b9e\u9a8c\u5ba4<\/a>\uff0c\u5b9e\u73b0\u601d\u60f3\u5171\u4eab\u3001\u8d44\u6e90\u5171\u4eab\u3001\u8de8\u5b66\u79d1\u4ea4\u53c9\uff0c\u5171\u540c\u4e3a\u590d\u6742\u7cfb\u7edf\u81ea\u52a8\u5efa\u6a21\u800c\u594b\u8fdb\u3002\u6b22\u8fce\u5bf9\u590d\u6742\u7cfb\u7edf\u81ea\u52a8\u5efa\u6a21\u9886\u57df\u6709\u70ed\u60c5\uff0c\u4e14\u8ba4\u53ef\u8fd9\u4e2a\u9886\u57df\u53d1\u5c55\u524d\u666f\u7684\u670b\u53cb\u4e00\u8d77\u6765\u5408\u4f5c\uff0c\u4fc3\u8fdb\u8fd9\u4e00\u9886\u57df\u7684\u5feb\u901f\u53d1\u5c55\u3002<\/span><\/strong><\/span><\/strong><\/span><\/strong><\/section>\n<\/section>\n
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\u5173\u952e\u8bcd\uff1a\u5947\u5f02\u503c\u5206\u89e3\uff0c\u8c31\u5206\u89e3<\/strong><\/strong><\/strong><\/strong><\/span><\/section>\n
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\u5218\u5b87\u5eb7<\/span> <\/span>| \u4f5c\u8005<\/span><\/p>\n

\u5f20\u6c5f\u3001\u738b\u5fd7\u9e4f<\/span> <\/span>| \u5ba1\u6821<\/span><\/span><\/p>\n<\/section>\n


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\u76ee\u5f55<\/strong><\/strong><\/p>\n

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1. <\/span><\/strong>\u76f4\u89c2\u89e3\u91ca<\/span><\/strong><\/section>\n

1.1 \u65cb\u8f6c\u3001\u5750\u6807\u7f29\u653e\u548c\u53cd\u5c04<\/span><\/p>\n

1.2 \u5947\u5f02\u503c\u4f5c\u4e3a\u692d\u5706\u6216\u692d\u7403\u4f53\u7684\u534a\u8f74<\/span><\/p>\n

1.3 U<\/span>\u548cV\u7684\u5217\u6784\u6210\u6807\u51c6\u6b63\u4ea4\u57fa<\/span><\/p>\n

1.4 \u4e0e\u56db\u4e2a\u57fa\u672c\u5b50\u7a7a\u95f4\u7684\u5173\u7cfb<\/span><\/p>\n

1.5 \u51e0\u4f55\u610f\u4e49<\/span><\/p>\n

2. \u6848\u4f8b<\/span><\/strong><\/section>\n
3. \u5947\u5f02\u503c\u5206\u89e3\uff08SVD\uff09\u548c\u8c31\u5206\u89e3<\/span><\/strong><\/section>\n

3.1 \u5947\u5f02\u503c\u3001\u5947\u5f02\u5411\u91cf\u53ca\u5176\u4e0eSVD\u7684\u5173\u7cfb<\/span><\/p>\n

3.2 \u4e0e\u7279\u5f81\u503c\u5206\u89e3\u7684\u5173\u7cfb<\/span><\/p>\n

4. \u5947\u5f02\u503c\u5206\u89e3\u7684\u5e94\u7528<\/span><\/strong><\/section>\n

4.1 \u4f2a\u9006<\/span><\/p>\n

4.2 \u6c42\u89e3\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b<\/span><\/p>\n

4.3 \u603b\u4f53\u6700\u5c0f\u4e8c\u4e58\u6700\u5c0f\u5316<\/span><\/p>\n

4.4 \u503c\u57df\u3001\u96f6\u7a7a\u95f4\u548c\u79e9<\/span><\/p>\n

4.5 \u4f4e\u79e9\u77e9\u9635\u8fd1\u4f3c<\/span><\/p>\n

4.6 \u53ef\u5206\u79bb\u6a21\u578b<\/span><\/p>\n

4.7 \u6700\u8fd1\u6b63\u4ea4\u77e9\u9635<\/span><\/p>\n

4.8 Kabsch\u7b97\u6cd5<\/span><\/p>\n

4.9 \u4fe1\u53f7\u5904\u7406<\/span><\/p>\n

4.10 \u5176\u4ed6\u4f8b\u5b50<\/span><\/p>\n

5. \u5b58\u5728\u6027\u8bc1\u660e<\/span><\/strong><\/section>\n

5.1 \u57fa\u4e8e\u8c31\u5b9a\u7406<\/span><\/p>\n

5.2 \u57fa\u4e8e\u53d8\u5206\u8868\u5f81\uff08variational characterization\uff09<\/span><\/p>\n

6. \u8ba1\u7b97\u5947\u5f02\u503c\u5206\u89e3\uff08SVD\uff09<\/span><\/strong><\/section>\n

6.1 \u5355\u8fb9\u96c5\u53ef\u6bd4\u7b97\u6cd5\uff08One-sided Jacobi algorithm\uff09<\/span><\/p>\n

6.2 \u53cc\u8fb9\u96c5\u53ef\u6bd4\u7b97\u6cd5\uff08Two-sided Jacobi algorithm\uff09<\/span><\/p>\n

6.3 \u6570\u503c\u65b9\u6cd5<\/span><\/p>\n

6.4 2 \u00d7 2 SVD\u7684\u89e3\u6790\u7ed3\u679c<\/span><\/p>\n

7. <\/span><\/strong>\u7b80\u5316\u5947\u5f02\u503c\u5206\u89e3<\/span><\/strong><\/section>\n

7.1 \u8584SVD (Thin SVD)<\/span><\/p>\n

7.2 \u7d27\u51d1SVD (Compact SVD)<\/span><\/p>\n

7.3 \u622a\u65adSVD (Truncated SVD)<\/span><\/p>\n

8. \u8303\u6570<\/span><\/strong><\/section>\n

8.1 Ky Fan\u8303\u6570<\/span><\/p>\n

8.2 \u5e0c\u5c14\u4f2f\u7279-\u65bd\u5bc6\u7279\u8303\u6570<\/span><\/p>\n

9. \u53d8\u4f53\u548c\u63a8\u5e7f<\/span><\/strong><\/section>\n

9.1 \u5c3a\u5ea6\u4e0d\u53d8\u5947\u5f02\u503c\u5206\u89e3<\/span><\/p>\n

9.2 \u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u4e0a\u7684\u6709\u754c\u7b97\u5b50<\/span><\/p>\n

9.3 \u5947\u5f02\u503c\u548c\u7d27\u7b97\u5b50<\/span><\/p>\n

10. \u5386\u53f2<\/span><\/strong><\/section>\n
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1. \u76f4\u89c2\u89e3\u91ca<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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1.1 \u65cb\u8f6c\u3001\u5750\u6807\u7f29\u653e\u548c\u53cd\u5c04<\/span><\/strong><\/span><\/h3>\n

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\u7279\u6b8a\u60c5\u51b5\u4e0b\uff0c\u5f53<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u662f<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u00d7<\/span><\/mo>m<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u5b9e\u65b9\u9635\u65f6\uff0c\u6211\u4eec\u4e5f\u53ef\u4ee5\u5c06\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u9009\u4e3a\u5b9e<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u00d7<\/span><\/mo>m<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u77e9\u9635\u3002\u6b64\u65f6\uff0c\u201c\u9149\u77e9\u9635\u201d\u548c\u201c\u6b63\u4ea4\u77e9\u9635\u201d\u5b9e\u9645\u4e0a\u662f\u4e00\u56de\u4e8b\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u8fd9\u4e24\u4e2a\u9149\u77e9\u9635\u548c\u5bf9\u89d2\u77e9\u9635<\/span>\uff08\u8fd9\u91cc\u7edf\u79f0\u4e3a<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff09<\/span>\u89e3\u8bfb\u4e3a\u7a7a\u95f4<\/span><\/sub><\/span><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u7ebf\u6027\u53d8\u6362<\/span>\uff08linear transformation\uff09<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>x<\/span><\/mi>\u21a6<\/span><\/mo>Ax<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u5176\u4e2d\uff0c\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4ee3\u8868\u7a7a\u95f4\u7684\u65cb\u8f6c<\/span>\uff08rotations\uff09<\/span>\u6216\u53cd\u5c04<\/span>\uff08reflection\uff09<\/span>\uff0c\u800c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5219\u8868\u793a\u5bf9\u6bcf\u4e2a\u5750\u6807<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>x<\/span><\/mi>i<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u6309\u56e0\u5b50<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>i<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u8fdb\u884c\u7684\u7f29\u653e\u53d8\u6362<\/span>\uff08scaling\uff09<\/span>\u3002\u8fd9\u6837\uff0c\u5947\u5f02\u503c\u5206\u89e3\u5c31\u628a<\/span><\/sub><\/span><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u4efb\u4f55\u7ebf\u6027\u53d8\u6362\u5206\u89e3\u6210\u4e86\u4e09\u4e2a\u51e0\u4f55\u53d8\u6362\u7684\u7ec4\u5408\uff1a\u5148\u65cb\u8f6c\u6216\u53cd\u5c04\uff08<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff09\uff0c\u7136\u540e\u9010\u5750\u6807\u7f29\u653e\uff08<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff09\uff0c\u6700\u540e\u518d\u65cb\u8f6c\u6216\u53cd\u5c04\uff08<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff09\u3002<\/span><\/p>\n


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\u7279\u522b\u5730\uff0c\u5982\u679c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u884c\u5217\u5f0f\u4e3a\u6b63\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u9700\u8981\u540c\u65f6\u5e26\u53cd\u5c04\u6216\u540c\u65f6\u90fd\u4e0d\u5e26\u53cd\u5c04\u3002\u82e5\u884c\u5217\u5f0f\u4e3a\u8d1f\uff0c\u5219\u53ea\u6709\u4e00\u4e2a\u4f1a\u5e26\u53cd\u5c04\u3002\u82e5\u884c\u5217\u5f0f\u4e3a\u96f6\uff0c\u6211\u4eec\u53ef\u4ee5\u968f\u610f\u9009\u62e9\u6bcf\u4e2a\u77e9\u9635\u7684\u7c7b\u578b\u3002<\/span><\/p>\n


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\u5f53<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u662f\u5b9e\u77e9\u9635\u4f46\u975e\u65b9\u9635\uff0c\u5373<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u00d7<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e14<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u2260<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u65f6<\/span>\uff08\u5982\u4e0b\u56fe\uff09<\/span>\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u89c6\u4e3a\u4ece<\/span><\/sub><\/span><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5230<\/span><\/sub><\/mjx-assistive-mml><\/mjx-container><\/span><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u7ebf\u6027\u53d8\u6362\u3002\u8fd9\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u9009\u62e9<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5206\u522b\u4e3a<\/span><\/sub><\/span><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/sub><\/span><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u65cb\u8f6c\/\u53cd\u5c04\uff1b\u800c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u9664\u4e86\u7f29\u653e\u524d<\/span><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>min<\/span><\/mo>{<\/span><\/mo>m<\/span><\/mi>,<\/span><\/mo>n<\/span><\/mi>}<\/span><\/mo><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e2a\u5750\u6807\u5916\uff0c\u8fd8\u4f1a\u7528\u96f6\u6269\u5c55\u5411\u91cf\u6216\u5220\u9664\u5c3e\u90e8\u5750\u6807\uff0c\u4ece\u800c\u5c06<\/span><\/sub><\/mjx-assistive-mml><\/mjx-container><\/span><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u8f6c\u6362<\/span>\u4e3a<\/span><\/sub><\/span><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002<\/span><\/p>\n


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1.2 \u5947\u5f02\u503c\u4f5c\u4e3a\u692d\u5706\u6216\u692d\u7403\u4f53\u7684\u534a\u8f74<\/span><\/strong><\/span><\/h3>\n

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\u5982\u4e0b\u56fe\u6240\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5947\u5f02\u503c\u7406\u89e3\u4e3a\u4e8c\u7ef4\u692d\u5706\u534a\u8f74\u7684\u957f\u5ea6\u3002\u8fd9\u4e00\u6982\u5ff5\u53ef\u4ee5\u63a8\u5e7f\u5230<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7ef4\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4<\/span>\uff08Euclidean space\uff09<\/span>\uff1a\u4efb\u4f55<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>n<\/span><\/mi>\u00d7<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u65b9\u9635\u7684\u5947\u5f02\u503c\u90fd\u53ef\u4ee5\u770b\u4f5c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7ef4\u692d\u7403\u4f53\u534a\u8f74\u7684\u957f\u5ea6\u3002\u540c\u7406\uff0c\u4efb\u4f55<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u00d7<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u77e9\u9635\u7684\u5947\u5f02\u503c\u53ef\u4ee5\u89c6\u4e3a<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7ef4\u7a7a\u95f4\u4e2d<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7ef4\u692d\u7403\u4f53\u534a\u8f74\u7684\u957f\u5ea6\uff0c\u6bd4\u5982\u4e09\u7ef4\u7a7a\u95f4\u4e2d<\/span>\uff08\u503e\u659c\u7684\uff09<\/span>\u4e8c\u7ef4\u5e73\u9762\u4e0a\u7684\u692d\u5706\u3002\u5947\u5f02\u503c\u7f16\u7801\u4e86\u534a\u8f74\u7684\u957f\u5ea6\uff0c\u800c\u5947\u5f02\u5411\u91cf\u5219\u7f16\u7801\u4e86\u65b9\u5411\u3002\u8be6\u89c1\u4e0b\u6587\uff1a<\/span><\/p>\n


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<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/h3>\n

<\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>1.3 U<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/strong><\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/strong><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u5217\u6784\u6210\u6807\u51c6\u6b63\u4ea4\u57fa<\/span><\/strong><\/span><\/h3>\n


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\u7531\u4e8e<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u90fd\u662f\u9149\u77e9\u9635\uff0c\u5b83\u4eec\u7684\u5217\u5206\u522b\u5f62\u6210\u4e00\u7ec4\u6807\u51c6\u6b63\u4ea4\u5411\u91cf<\/span>\uff08orthonormal vectors \u6a21\u4e3a1\u4e14\u5f7c\u6b64\u6b63\u4ea4\u7684\u5411\u91cf\uff09<\/span>\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u89c6\u4e3a\u57fa\u5411\u91cf<\/span>\uff08basis vectors \u5f20\u6210\u5411\u91cf\u7a7a\u95f4\u7684\u4e00\u7ec4\u7ebf\u6027\u65e0\u5173\u5411\u91cf\uff09<\/span>\u3002\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u628a\u57fa\u5411\u91cf<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>i<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u6620\u5c04\u5230\u62c9\u4f38\u540e\u7684\u5355\u4f4d\u5411\u91cf<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>i<\/span><\/sub><\/mi><\/msub>U<\/span><\/mi><\/mrow>i<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e0a\u3002\u6839\u636e\u9149\u77e9\u9635\u7684\u5b9a\u4e49\uff0c\u5b83\u4eec\u7684\u5171\u8f6d\u8f6c\u7f6e<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e5f\u5177\u6709\u76f8\u540c\u6027\u8d28\uff0c\u53ea\u662f\u5931\u53bb\u4e86\u5947\u5f02\u503c\u4f5c\u4e3a\u62c9\u4f38\u7684\u51e0\u4f55\u89e3\u91ca\u3002\u7b80\u8a00\u4e4b\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3001<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3001<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u5217\u90fd\u6784\u6210\u6807\u51c6\u6b63\u4ea4\u57fa<\/span>\uff08orthonormal bases \u6a21\u4e3a1\u4e14\u6b63\u4ea4\u7684\u57fa\u5411\u91cf\u96c6\u5408\uff09<\/span>\u3002<\/span><\/p>\n


<\/span><\/p>\n

\u5f53<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u662f\u534a\u6b63\u5b9a\u5384\u7c73\u77e9\u9635<\/span>\uff08positive-semidefinite Hermitian matrix\uff09<\/span>\u65f6\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u90fd\u7b49\u540c\u4e8e\u7528\u4e8e\u5bf9\u89d2\u5316<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u9149\u77e9\u9635\u3002\u7136\u800c\uff0c\u5982\u679c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e0d\u662f\u534a\u6b63\u5b9a\u5384\u7c73\u77e9\u9635\u4f46\u4ecd\u53ef\u5bf9\u89d2\u5316\u7684\u65f6\u5019\uff0c\u90a3\u4e48\u5176\u7279\u5f81\u5206\u89e3<\/span>\uff08\u77e9\u9635\u5206\u89e3\u4e3a\u7279\u5f81\u503c\u4e0e\u7279\u5f81\u5411\u91cf\uff09<\/span>\u548c\u5947\u5f02\u503c\u5206\u89e3\u5c31\u4f1a\u6709\u6240\u4e0d\u540c\u3002<\/span><\/p>\n


<\/span><\/p>\n

1.4 \u4e0e\u56db\u4e2a\u57fa\u672c\u5b50\u7a7a\u95f4\u7684\u5173\u7cfb\uff1a<\/span><\/strong><\/span><\/h3>\n

<\/span><\/section>\n
    \n
  • \n
    <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u524d<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>r<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5217\u6784\u6210\u4e86<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u5217\u7a7a\u95f4<\/span>\uff08\u77e9\u9635\u5217\u5411\u91cf\u5f20\u6210\u7684\u7a7a\u95f4\uff09<\/span>\u7684\u57fa\u3002<\/span><\/section>\n<\/li>\n
  • \n
    <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u540e<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u2212<\/span><\/mo>r<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5217\u5219\u6784\u6210<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u96f6\u7a7a\u95f4<\/span>\uff08\u77e9\u9635\u6620\u5c04\u4e2d\u88ab\u6620\u5c04\u5230\u96f6\u7684\u6240\u6709\u5411\u91cf\u96c6\u5408\uff09<\/span>\u7684\u57fa\u3002<\/span><\/section>\n<\/li>\n
  • \n
    <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u524d<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>r<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5217\u6784\u6210<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5217\u7a7a\u95f4\u7684\u57fa<\/span>\uff08\u5728\u5b9e\u6570\u60c5\u51b5\u4e0b\u5373\u4e3a<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u884c\u7a7a\u95f4\uff08\u77e9\u9635\u884c\u5411\u91cf\u5f20\u6210\u7684\u7a7a\u95f4\uff09\u7684\u57fa\uff09<\/span>\u3002<\/span><\/section>\n<\/li>\n
  • \n

    <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u540e<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>n<\/span><\/mi>\u2212<\/span><\/mo>r<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5217\u6784\u6210<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u96f6\u7a7a\u95f4\u7684\u57fa\u3002<\/span><\/p>\n<\/li>\n<\/ul>\n


    <\/span><\/h3>\n

    1.5 \u51e0\u4f55\u610f\u4e49\uff1a<\/span><\/strong><\/span><\/h3>\n

    <\/span><\/section>\n
    \u7531\u4e8e<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u662f\u9149\u77e9\u9635\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u5217<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub>,<\/span><\/mo>\u2026<\/span><\/mo>,<\/span><\/mo>U<\/span><\/mi><\/mrow>m<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u6784\u6210<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>K<\/span><\/mi>m<\/span><\/sup><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u6807\u51c6\u6b63\u4ea4\u57fa\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u5217<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub>,<\/span><\/mo>\u2026<\/span><\/mo>,<\/span><\/mo>V<\/span><\/mi><\/mrow>n<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u6784\u6210<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>K<\/span><\/mi>n<\/span><\/sup><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u6807\u51c6\u6b63\u4ea4\u57fa<\/span>\uff08\u57fa\u4e8e\u8fd9\u4e9b\u7a7a\u95f4\u7684\u6807\u91cf\u79ef\uff08scalar products \u4e24\u4e2a\u5411\u91cf\u95f4\u7684\u5185\u79ef\uff09\uff09<\/span>\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    \u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\u4e3a\uff1a<\/span><\/section>\n

    <\/span><\/section>\n
    <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-tstrut><\/mjx-mtd><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mi><\/mjx-tstrut><\/mjx-mtd><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-texatom><\/mjx-mo><\/mjx-mrow><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

    <\/p>\n

    <\/mi><\/mtd><\/mtr><\/mtable><\/mrow><\/mo><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

    <\/span><\/section>\n
    \u5219\u8be5\u6620\u5c04\u5728\u8fd9\u4e9b\u6807\u51c6\u6b63\u4ea4\u57fa\u4e0b\u6709\u4e00\u4e2a\u7b80\u6d01\u7684\u63cf\u8ff0\uff1a<\/span><\/section>\n

    <\/span><\/section>\n
    <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

    <\/p>\n

    <\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

    <\/span><\/section>\n
    \u5176\u4e2d<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>i<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u662f<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u7b2c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>i<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e2a\u5bf9\u89d2\u5143\u7d20\uff0c\u5f53<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>i<\/span><\/mi>><\/span><\/mo>min<\/span><\/mo>(<\/span><\/mo>m<\/span><\/mi>,<\/span><\/mo>n<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u65f6\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>T<\/span><\/mi>(<\/span><\/mo>V<\/span><\/mi><\/mrow>i<\/span><\/sub><\/mi><\/msub>)<\/span><\/mo>=<\/span><\/mo>0<\/span><\/mn><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002<\/span><\/section>\n

    \u6240\u8c13\u7684SVD\u5b9a\u7406\u7684\u51e0\u4f55\u542b\u4e49\u53ef\u4ee5\u6982\u62ec\u4e3a\uff1a\u5bf9\u4e8e\u6bcf\u4e2a\u7ebf\u6027\u6620\u5c04<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>T<\/span><\/mi>:<\/span><\/mo>K<\/span><\/mi>n<\/span><\/sup><\/mi><\/msup>\u2192<\/span><\/mo>K<\/span><\/mi>m<\/span><\/sup><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u6211\u4eec\u80fd\u627e\u5230<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>K<\/span><\/mi>n<\/span><\/sup><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>K<\/span><\/mi>m<\/span><\/sup><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u6807\u51c6\u6b63\u4ea4\u57fa\uff0c\u4f7f<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>T<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5c06<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>K<\/span><\/mi>n<\/span><\/sup><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u7b2c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>i<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e2a\u57fa\u5411\u91cf\u6620\u5c04\u5230<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>K<\/span><\/mi>m<\/span><\/sup><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u7b2c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>i<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e2a\u57fa\u5411\u91cf\u7684\u975e\u8d1f\u500d\u6570\uff0c\u5e76\u5c06\u5269\u4f59\u57fa\u5411\u91cf\u6620\u5c04\u5230\u96f6\u3002\u5728\u8fd9\u4e9b\u57fa\u4e0b\uff0c\u6620\u5c04<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>T<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u8868\u73b0\u4e3a\u4e00\u4e2a\u5177\u6709\u975e\u8d1f\u5b9e\u6570\u5bf9\u89d2\u5143\u7d20\u7684\u5bf9\u89d2\u77e9\u9635\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    \u4e3a\u4e86\u66f4\u76f4\u89c2\u5730\u7406\u89e3\u5947\u5f02\u503c\u548cSVD\u5206\u89e3<\/span>\uff08\u81f3\u5c11\u5728\u5b9e\u5411\u91cf\u7a7a\u95f4\u4e2d\uff09<\/span>\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>R<\/span><\/mi><\/mrow>n<\/span><\/sup><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e2d\u534a\u5f84\u4e3a1\u7684\u7403\u9762<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>S<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u7ebf\u6027\u6620\u5c04<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>T<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5c06\u8fd9\u4e2a\u7403\u9762\u6620\u5c04\u5230<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>R<\/span><\/mi><\/mrow>m<\/span><\/sup><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e2d\u7684\u4e00\u4e2a\u692d\u7403\u4f53\u3002\u975e\u96f6\u5947\u5f02\u503c\u5c31\u662f\u8fd9\u4e2a\u692d\u7403\u4f53\u534a\u8f74<\/span>\uff08semi-axes\uff09<\/span>\u7684\u957f\u5ea6\u3002\u7279\u522b\u5730\uff0c\u5f53<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>n<\/span><\/mi>=<\/span><\/mo>m<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e14\u6240\u6709\u5947\u5f02\u503c\u90fd\u4e0d\u540c\u4e14\u975e\u96f6\u65f6\uff0c\u7ebf\u6027\u6620\u5c04<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>T<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684SVD\u53ef\u4ee5\u5206\u89e3\u4e3a\u4e09\u4e2a\u8fde\u7eed\u6b65\u9aa4\uff1a<\/span><\/section>\n

    <\/span><\/section>\n
    1. \u8003\u8651\u692d\u7403\u4f53<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>T<\/span><\/mi>(<\/span><\/mo>S<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u53ca\u5176\u8f74\uff0c\u7136\u540e\u627e\u51fa<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>R<\/span><\/mi><\/mrow>n<\/span><\/sup><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e2d\u88ab<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>T<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u6620\u5c04\u5230\u8fd9\u4e9b\u8f74\u4e0a\u7684\u65b9\u5411\u3002\u8fd9\u4e9b\u65b9\u5411\u6070\u597d\u76f8\u4e92\u6b63\u4ea4\u3002\u9996\u5148\u5e94\u7528\u7b49\u8ddd\u53d8\u6362<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u5c06\u8fd9\u4e9b\u65b9\u5411\u6620\u5c04\u5230<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>R<\/span><\/mi><\/mrow>n<\/span><\/sup><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u5750\u6807\u8f74\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    2. \u5e94\u7528\u6cbf\u5750\u6807\u8f74\u5bf9\u89d2\u5316\u7684\u81ea\u540c\u6001<\/span>\uff08endomorphism \u6620\u5c04\u81ea\u8eab\u5230\u81ea\u8eab\u7684\u7ebf\u6027\u53d8\u6362\uff09<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>D<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u5728\u6bcf\u4e2a\u65b9\u5411\u4e0a\u8fdb\u884c\u62c9\u4f38\u6216\u6536\u7f29\uff0c\u4f7f\u7528<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>T<\/span><\/mi>(<\/span><\/mo>S<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u534a\u8f74\u957f\u5ea6\u4f5c\u4e3a\u62c9\u4f38\u7cfb\u6570\u3002\u7ec4\u5408<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>D<\/span><\/mi><\/mrow>\u2218<\/span><\/mo>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5c06\u5355\u4f4d\u7403\u9762\u6620\u5c04\u5230\u4e0e<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>T<\/span><\/mi>(<\/span><\/mo>S<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7b49\u8ddd\u7684\u692d\u7403\u4f53\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    3. \u6700\u540e\uff0c\u5bf9\u8fd9\u4e2a\u692d\u7403\u4f53\u5e94\u7528\u7b49\u8ddd\u53d8\u6362<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4ee5\u5f97\u5230<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>T<\/span><\/mi>(<\/span><\/mo>S<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    \u53ef\u4ee5\u9a8c\u8bc1\uff0c\u7ec4\u5408<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>\u2218<\/span><\/mo>D<\/span><\/mi><\/mrow>\u2218<\/span><\/mo>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e0e<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>T<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e00\u81f4\u3002<\/span><\/section>\n


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    <\/svg><\/section>\n<\/section>\n<\/section>\n
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    2. \u6848\u4f8b<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

    \n

    <\/section>\n

    <\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


    <\/p>\n

    \u8ba9\u6211\u4eec\u6765\u770b\u4e00\u4e2a<\/span><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>4<\/span><\/mn>\u00d7<\/span><\/mo>5<\/span><\/mn><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u77e9\u9635\u7684\u4f8b\u5b50\uff1a<\/span><\/section>\n

    <\/span><\/section>\n
    <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-beg><\/mjx-c><\/mjx-ext><\/mjx-c><\/mjx-end><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-c><\/mjx-beg><\/mjx-c><\/mjx-ext><\/mjx-c><\/mjx-end><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><\/mjx-mrow><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n
    <\/section>\n
    <\/mo><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

    <\/span><\/section>\n
    \u6211\u4eec\u53ef\u4ee5\u901a\u8fc7<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>\u03a3<\/span><\/mi><\/mrow>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5bf9\u8be5\u77e9\u9635\u8fdb\u884c\u5947\u5f02\u503c\u5206\u89e3\uff1a<\/span><\/section>\n

    <\/span><\/section>\n
    <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-beg><\/mjx-c><\/mjx-ext><\/mjx-c><\/mjx-end><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-c><\/mjx-beg><\/mjx-c><\/mjx-ext><\/mjx-c><\/mjx-end><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><\/mjx-mrow><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

    <\/p>\n

    <\/mo><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

    <\/span><\/section>\n
    \u6ce8\u610f\uff0c\u7f29\u653e\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5728\u5bf9\u89d2\u7ebf\u4ee5\u5916\u7684\u5143\u7d20\u90fd\u4e3a\u96f6<\/span>\uff08\u7528\u7070\u8272\u659c\u4f53\u8868\u793a\uff09<\/span>\uff0c\u4e14\u6709\u4e00\u4e2a\u5bf9\u89d2\u7ebf\u5143\u7d20\u4e3a\u96f6\uff08\u7528\u7ea2\u8272\u7c97\u4f53\u8868\u793a\uff09\u3002\u7531\u4e8e<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u90fd\u662f\u9149\u77e9\u9635\uff0c\u5c06\u5b83\u4eec\u5206\u522b\u4e0e\u5176\u5171\u8f6d\u8f6c\u7f6e\u76f8\u4e58\u4f1a\u5f97\u5230\u5355\u4f4d\u77e9\u9635<\/span>\uff08identity matrices \u5bf9\u89d2\u7ebf\u4e3a1\uff0c\u5176\u4f59\u4e3a0\u7684\u77e9\u9635\uff09<\/span>\u3002\u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u90fd\u662f\u5b9e\u503c\u77e9\u9635\uff0c\u56e0\u6b64\u5b83\u4eec\u90fd\u662f\u6b63\u4ea4\u77e9\u9635<\/span>\uff08\u5217\u5411\u91cf\u4e92\u76f8\u6b63\u4ea4\u4e14\u5355\u4f4d\u957f\u5ea6\uff09<\/span>\u3002\u6211\u4eec\u53ef\u4ee5\u9a8c\u8bc1\uff1a<\/span><\/section>\n

    <\/span><\/section>\n
    <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-beg><\/mjx-c><\/mjx-ext><\/mjx-c><\/mjx-end><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-c><\/mjx-beg><\/mjx-c><\/mjx-ext><\/mjx-c><\/mjx-end><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><\/mjx-mrow><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

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    <\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

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    \u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u8fd9\u4e2a\u5947\u5f02\u503c\u5206\u89e3\u5e76\u975e\u552f\u4e00\u3002\u6211\u4eec\u8fd8\u53ef\u4ee5\u9009\u62e9\u53e6\u4e00\u4e2a<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u4f7f\u5f97\uff1a<\/span><\/section>\n

    <\/span><\/section>\n
    <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-beg><\/mjx-c><\/mjx-ext><\/mjx-c><\/mjx-end><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mo><\/mjx-surd><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><\/mjx-texatom><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mo><\/mjx-surd><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><\/mjx-texatom><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mo><\/mjx-surd><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><\/mjx-texatom><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mo><\/mjx-surd><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><\/mjx-texatom><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-surd><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><\/mjx-texatom><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-surd><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><\/mjx-texatom><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mo><\/mjx-surd><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><\/mjx-texatom><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mo><\/mjx-surd><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><\/mjx-texatom><\/mjx-mstyle><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-c><\/mjx-beg><\/mjx-c><\/mjx-ext><\/mjx-c><\/mjx-end><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><\/mjx-mrow><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

    <\/p>\n

    <\/mo><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

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    \u8fd9\u540c\u6837\u662f\u4e00\u4e2a\u6709\u6548\u7684\u5947\u5f02\u503c\u5206\u89e3\u3002<\/span><\/section>\n

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    3. \u5947\u5f02\u503c\u5206\u89e3\uff08SVD\uff09\u548c\u8c31\u5206\u89e3<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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    <\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


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    3.1 \u5947\u5f02\u503c\u3001\u5947\u5f02\u5411\u91cf\u53ca\u5176\u4e0eSVD\u7684\u5173\u7cfb<\/span><\/strong><\/span><\/h3>\n

    <\/span><\/section>\n
    \u975e\u8d1f\u5b9e\u6570 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u79f0\u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5947\u5f02\u503c\uff0c\u5f53\u4e14\u4ec5\u5f53\u5b58\u5728 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>K<\/span><\/mi>m<\/span><\/sup><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e2d\u7684\u5355\u4f4d\u957f\u5ea6\u5411\u91cf <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>u<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>K<\/span><\/mi>n<\/span><\/sup><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e2d\u7684\u5355\u4f4d\u957f\u5ea6\u5411\u91cf <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>v<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u6ee1\u8db3\uff1a<\/span><\/section>\n

    <\/span><\/section>\n
    <\/mjx-c><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-tstrut><\/mjx-mtd><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-tstrut><\/mjx-mtd><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

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    <\/mo><\/mtd><\/mtr><\/mtable><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

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    \u6211\u4eec\u79f0\u5411\u91cf <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>u<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>v<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5206\u522b\u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5de6\u5947\u5f02\u5411\u91cf\u548c\u53f3\u5947\u5f02\u5411\u91cf\u3002<\/span><\/section>\n

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    \u5728\u5947\u5f02\u503c\u5206\u89e3 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>=<\/span><\/mo>U<\/span><\/mi><\/mrow>\u03a3<\/span><\/mi><\/mrow>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> <\/span><\/sup>\u4e2d\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5bf9\u89d2\u7ebf\u5143\u7d20\u5373\u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5947\u5f02\u503c\u3002<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u524d <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>p<\/span><\/mi>=<\/span><\/mo>min<\/span><\/mo>(<\/span><\/mo>m<\/span><\/mi>,<\/span><\/mo>n<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5217\u5206\u522b\u5bf9\u5e94\u5404\u5947\u5f02\u503c\u7684\u5de6\u5947\u5f02\u5411\u91cf\u548c\u53f3\u5947\u5f02\u5411\u91cf\u3002\u56e0\u6b64\uff0c\u4e0a\u8ff0\u5b9a\u7406\u8868\u660e\uff1a<\/span><\/p>\n

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      \u4e00\u4e2a <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u00d7<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u77e9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u6700\u591a\u6709 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>p<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e2a\u4e0d\u540c\u7684\u5947\u5f02\u503c\u3002<\/span><\/section>\n<\/li>\n
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      \u6211\u4eec\u603b\u80fd\u5728 \u2060<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>K<\/span><\/mi>m<\/span><\/sup><\/mi><\/msup><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u2060 \u7a7a\u95f4\u4e2d\u627e\u5230\u4e00\u7ec4\u9149\u57fa<\/span>\uff08\u5373\u4e00\u7ec4\u57fa\u5411\u91cf\uff0c\u5b83\u4eec\u6784\u6210\u7684\u77e9\u9635\u662f\u9149\u77e9\u9635\uff09<\/span> <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>\u2060<\/span><\/mo><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u2060\uff0c\u5176\u4e2d\u90e8\u5206\u57fa\u5411\u91cf\uff0c\u5f20\u6210\u4e86\u77e9\u9635 <\/span><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u2060<\/span><\/mo><\/mrow>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u2060 \u4e2d\u6bcf\u4e2a\u5947\u5f02\u503c\u5bf9\u5e94\u7684\u5de6\u5947\u5f02\u5411\u91cf\u3002<\/span><\/section>\n<\/li>\n
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      \u540c\u6837\u5730\uff0c\u5728 <\/span><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u2060<\/span><\/mo><\/mrow>K<\/span><\/mi>n<\/span><\/sup><\/mi><\/msup><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u2060 \u7a7a\u95f4\u4e2d\u4e5f\u5b58\u5728\u4e00\u7ec4\u9149\u57fa <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>\u2060<\/span><\/mo><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u2060\uff0c\u5176\u4e2d\u90e8\u5206\u57fa\u5411\u91cf\uff0c\u5f20\u6210\u4e86\u77e9\u9635 <\/span>M<\/span><\/mi><\/mrow>\u2060 <\/span><\/mo><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e2d\u6bcf\u4e2a\u5947\u5f02\u503c\u5bf9\u5e94\u7684\u53f3\u5947\u5f02\u5411\u91cf\u3002<\/span><\/p>\n<\/li>\n<\/ul>\n


      <\/span><\/section>\n
      \u5f53\u6211\u4eec\u80fd\u627e\u5230\u4e24\u4e2a\u7ebf\u6027\u72ec\u7acb\u7684\u5de6<\/span>\uff08\u6216\u53f3\uff09<\/span>\u5947\u5f02\u5411\u91cf\u65f6\uff0c\u6211\u4eec\u79f0\u8be5\u5947\u5f02\u503c\u4e3a\u7b80\u5e76<\/span>\uff08degenerate\uff09<\/span>\u7684\u3002\u5982\u679c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>u<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>u<\/span><\/mi><\/mrow>2<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u5bf9\u5e94\u5947\u5f02\u503c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u4e24\u4e2a\u5de6\u5947\u5f02\u5411\u91cf\uff0c\u90a3\u4e48\u8fd9\u4e24\u4e2a\u5411\u91cf\u7684\u4efb\u4f55\u5f52\u4e00\u5316\u7ebf\u6027\u7ec4\u5408\u4e5f\u662f\u5bf9\u5e94\u5947\u5f02\u503c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5de6\u5947\u5f02\u5411\u91cf\u3002\u53f3\u5947\u5f02\u5411\u91cf\u4e5f\u6709\u7c7b\u4f3c\u6027\u8d28\u3002\u72ec\u7acb\u7684\u5de6\u5947\u5f02\u5411\u91cf\u548c\u53f3\u5947\u5f02\u5411\u91cf\u6570\u91cf\u76f8\u540c\uff0c\u5b83\u4eec\u5206\u522b\u51fa\u73b0\u5728 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5bf9\u5e94\u5217\u4e2d\uff0c\u8fd9\u4e9b\u5217\u5bf9\u5e94\u7740 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e2d\u503c\u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5bf9\u89d2\u5143\u7d20\u3002<\/span><\/section>\n

      <\/span><\/section>\n
      \u7279\u522b\u5730\uff0c\u5947\u5f02\u503c\u4e3a0\u7684\u5de6\u5947\u5f02\u5411\u91cf\u548c\u53f3\u5947\u5f02\u5411\u91cf\u5206\u522b\u5305\u62ec <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u4f59\u6838<\/span>\uff08cokernel \u672a\u88ab\u77e9\u9635\u8986\u76d6\u7684\u8f93\u51fa\u7a7a\u95f4\uff09<\/span>\u548c\u6838<\/span>\uff08kernel \u77e9\u9635\u6620\u5c04\u4e3a\u96f6\u7684\u8f93\u5165\u7a7a\u95f4\uff09<\/span>\u4e2d\u7684\u6240\u6709\u5355\u4f4d\u5411\u91cf\u3002\u6839\u636e\u79e9-\u96f6\u5316\u5ea6\u5b9a\u7406<\/span>\uff08rank\u2013nullity theorem\uff09<\/span>\uff0c\u5982\u679c <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u2260<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u5b83\u4eec\u7684\u7ef4\u6570\u4e0d\u53ef\u80fd\u76f8\u540c\u3002\u5373\u4f7f\u6240\u6709\u5947\u5f02\u503c\u90fd\u975e\u96f6\uff0c\u5f53 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>><\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u65f6\uff0c\u4f59\u6838\u662f\u975e\u5e73\u51e1<\/span>\uff08nontrivial\uff09<\/span>\u7684\uff0c\u6b64\u65f6 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7528\u4f59\u6838\u4e2d\u7684 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u2212<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e2a\u6b63\u4ea4\u5411\u91cf\u586b\u5145\u3002\u76f8\u53cd\uff0c\u5f53 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi><<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u65f6\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7531\u6838\u4e2d\u7684 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>n<\/span><\/mi>\u2212<\/span><\/mo>m<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e2a\u6b63\u4ea4\u5411\u91cf\u586b\u5145\u3002\u7136\u800c\uff0c\u5982\u679c\u5b58\u57280\u7684\u5947\u5f02\u503c\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u6216 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u989d\u5916\u5217\u5df2\u7ecf\u4f5c\u4e3a\u5de6\u5947\u5f02\u5411\u91cf\u6216\u53f3\u5947\u5f02\u5411\u91cf\u51fa\u73b0\u3002<\/span><\/section>\n

      <\/span><\/section>\n
      \u975e\u7b80\u5e76\u5947\u5f02\u503c\u603b\u6709\u552f\u4e00\u7684\u5de6\u5947\u5f02\u5411\u91cf\u548c\u53f3\u5947\u5f02\u5411\u91cf\uff0c\u53ea\u80fd\u4e58\u4ee5\u5355\u4f4d\u76f8\u4f4d\u56e0\u5b50<\/span>\uff08unit-phase factor\uff09<\/span> <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>e<\/span><\/mi>i<\/span><\/sup><\/mi>\u03c6<\/span><\/sup><\/mi><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff08\u5b9e\u6570\u60c5\u51b5\u4e0b\uff0c\u53ea\u80fd\u4e58\u4ee5\u6b63\u8d1f\u53f7\uff09<\/span>\u3002\u56e0\u6b64\uff0c\u5982\u679c\u65b9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u6240\u6709\u5947\u5f02\u503c\u90fd\u662f\u975e\u7b80\u5e76\u4e14\u975e\u96f6\u7684\uff0c\u90a3\u4e48\u5b83\u7684\u5947\u5f02\u503c\u5206\u89e3\u662f\u552f\u4e00\u7684\uff0c\u53ea\u80fd\u5bf9 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u4e00\u5217\u4e58\u4ee5\u5355\u4f4d\u76f8\u4f4d\u56e0\u5b50\uff0c\u540c\u65f6\u5bf9 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u76f8\u5e94\u5217\u4e58\u4ee5\u76f8\u540c\u7684\u5355\u4f4d\u76f8\u4f4d\u56e0\u5b50\u3002\u603b\u7684\u6765\u8bf4\uff0cSVD \u5728\u5bf9 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5217\u5411\u91cf<\/span>\uff08\u8fd9\u4e9b\u5217\u5411\u91cf\u5f20\u6210\u6bcf\u4e2a\u5947\u5f02\u503c\u7684\u5b50\u7a7a\u95f4\uff09<\/span>\u7edf\u4e00\u5e94\u7528\u4efb\u610f\u9149\u53d8\u6362\uff0c\u4ee5\u53ca\u5bf9 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5411\u91cf<\/span>\uff08\u8fd9\u4e9b\u5411\u91cf\u5206\u522b\u5f20\u6210 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u6838\u548c\u4f59\u6838\uff09<\/span>\u5e94\u7528\u4efb\u610f\u9149\u53d8\u6362\u7684\u60c5\u51b5\u4e0b\u662f\u552f\u4e00\u7684\u3002<\/span><\/section>\n

      <\/span><\/section>\n

      3.2 \u4e0e\u7279\u5f81\u503c\u5206\u89e3\u7684\u5173\u7cfb<\/span><\/strong><\/span><\/h3>\n

      <\/span><\/section>\n
      \u5947\u5f02\u503c\u5206\u89e3\u5177\u6709\u5e7f\u6cdb\u9002\u7528\u6027\uff0c\u53ef\u7528\u4e8e\u4efb\u4f55 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u00d7<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u77e9\u9635\uff0c\u800c\u7279\u5f81\u503c\u5206\u89e3<\/span>\uff08eigenvalue decomposition\uff09<\/span>\u4ec5\u9650\u4e8e\u53ef\u5bf9\u89d2\u5316\u7684\u65b9\u9635\u3002\u5c3d\u7ba1\u5982\u6b64\uff0c\u8fd9\u4e24\u79cd\u5206\u89e3\u65b9\u6cd5\u4ecd\u6709\u5bc6\u5207\u8054\u7cfb\u3002<\/span><\/section>\n

      <\/span><\/section>\n
      \u8bbe <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684SVD\u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>=<\/span><\/mo>U<\/span><\/mi><\/mrow>\u03a3<\/span><\/mi><\/mrow>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u5219\u4ee5\u4e0b\u4e24\u4e2a\u7b49\u5f0f\u6210\u7acb\uff1a<\/span><\/section>\n

      <\/span><\/section>\n
      <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-tstrut><\/mjx-mtd><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-tstrut><\/mjx-mtd><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/msup><\/mtd><\/mtr><\/mtable><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

      <\/p>\n

      <\/mo><\/mtd><\/mtr><\/mtable><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

      <\/span><\/section>\n

      \u7b49\u5f0f\u53f3\u4fa7\u63cf\u8ff0\u4e86\u5de6\u4fa7\u7684\u7279\u5f81\u503c\u5206\u89e3\u3002\u7531\u6b64\u53ef\u5f97\uff1a<\/span><\/p>\n

        \n
      • \n
        <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5217<\/span>\uff08\u5373\u53f3\u5947\u5f02\u5411\u91cf\uff09<\/span>\u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u7279\u5f81\u5411\u91cf<\/span>\uff08eigenvectors \u77e9\u9635\u53d8\u6362\u4e0d\u6539\u53d8\u65b9\u5411\u7684\u5411\u91cf\uff09<\/span>\u3002<\/span><\/section>\n<\/li>\n
      • \n
        <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5217<\/span>\uff08\u5373\u5de6\u5947\u5f02\u5411\u91cf\uff09<\/span>\u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u7279\u5f81\u5411\u91cf\u3002<\/span><\/section>\n<\/li>\n
      • \n

        <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u975e\u96f6\u5143\u7d20<\/span>\uff08\u5373\u975e\u96f6\u5947\u5f02\u503c\uff09<\/span>\u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u6216 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u975e\u96f6\u7279\u5f81\u503c<\/span>\uff08\u77e9\u9635\u53d8\u6362\u540e\u7f29\u653e\u7684\u56e0\u5b50\uff09<\/span>\u7684\u5e73\u65b9\u6839\u3002<\/span><\/p>\n<\/li>\n<\/ul>\n


        <\/span><\/section>\n
        \u5f53 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e3a\u6b63\u89c4\u77e9\u9635<\/span>\uff08normal matrix \u4e0e\u5176\u5171\u8f6d\u8f6c\u7f6e\u53ef\u4ea4\u6362\u7684\u77e9\u9635\uff09<\/span>\u65f6\uff0c\u6839\u636e\u8c31\u5b9a\u7406<\/span>\uff08spectral theorem\uff09<\/span>\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u7279\u5f81\u5411\u91cf\u7684\u57fa\u5bf9\u5176\u8fdb\u884c\u9149\u5bf9\u89d2\u5316\uff0c\u5f97\u5230\u5206\u89e3 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>=<\/span><\/mo>U<\/span><\/mi><\/mrow>D<\/span><\/mi><\/mrow>U<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u5176\u4e2d <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u9149\u77e9\u9635\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>D<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u5bf9\u89d2\u7ebf\u4e0a\u6709\u590d\u6570\u5143\u7d20 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>i<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5bf9\u89d2\u77e9\u9635\u3002\u82e5 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e3a\u534a\u6b63\u5b9a<\/span>\uff08positive semi-definite \u6240\u6709\u7279\u5f81\u503c\u975e\u8d1f\u7684\u77e9\u9635\uff09<\/span>\u77e9\u9635\uff0c\u5219 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>i<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e3a\u975e\u8d1f\u5b9e\u6570\uff0c\u6b64\u65f6\u5206\u89e3 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>=<\/span><\/mo>U<\/span><\/mi><\/mrow>D<\/span><\/mi><\/mrow>U<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e5f\u662f\u4e00\u4e2a\u5947\u5f02\u503c\u5206\u89e3\u3002\u5426\u5219\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u6bcf\u4e2a <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>i<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u76f8\u4f4d <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi>e<\/span><\/mi>i<\/span><\/sup><\/mi>\u03c6<\/span><\/sup><\/mi><\/mrow><\/mi><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u79fb\u5230\u76f8\u5e94\u7684 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>i<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u6216 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>i<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e2d\uff0c\u4ece\u800c\u91cd\u65b0\u8868\u793a\u4e3aSVD\u5f62\u5f0f\u3002SVD\u4e0e\u975e\u6b63\u89c4\u77e9\u9635\u7684\u8054\u7cfb\u4e3b\u8981\u4f53\u73b0\u5728\u6781\u5206\u89e3\u5b9a\u7406\uff1a<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>=<\/span><\/mo>S<\/span><\/mi><\/mrow>R<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u5176\u4e2d <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>S<\/span><\/mi><\/mrow>=<\/span><\/mo>U<\/span><\/mi><\/mrow>\u03a3<\/span><\/mi><\/mrow>U<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u534a\u6b63\u5b9a\u4e14\u6b63\u89c4\u7684\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>R<\/span><\/mi><\/mrow>=<\/span><\/mo>U<\/span><\/mi><\/mrow>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u9149\u7684\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u56e0\u6b64\uff0c\u9664\u534a\u6b63\u5b9a\u77e9\u9635\u5916\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u7279\u5f81\u503c\u5206\u89e3\u548cSVD\u867d\u6709\u5173\u8054\uff0c\u4f46\u5e76\u4e0d\u76f8\u540c\uff1a\u7279\u5f81\u503c\u5206\u89e3\u5f62\u5982 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>=<\/span><\/mo>U<\/span><\/mi><\/mrow>D<\/span><\/mi><\/mrow>U<\/span><\/mi><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>–<\/span><\/sup><\/span><\/mo>1<\/span><\/sup><\/span><\/mn><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/mn><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u5176\u4e2d <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e0d\u4e00\u5b9a\u662f\u9149\u7684\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>D<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e0d\u4e00\u5b9a\u662f\u534a\u6b63\u5b9a\u7684\uff1b\u800cSVD\u5f62\u5982 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>=<\/span><\/mo>U<\/span><\/mi><\/mrow>\u03a3<\/span><\/mi><\/mrow>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u5176\u4e2d <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u5bf9\u89d2\u7684\u4e14\u534a\u6b63\u5b9a\u7684\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u9149\u77e9\u9635\uff0c\u5b83\u4eec\u9664\u4e86\u901a\u8fc7\u77e9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5916\u4e0d\u4e00\u5b9a\u6709\u5173\u8054\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u53ea\u6709\u975e\u4e8f\u635f<\/span>\uff08non-defective \u77e9\u9635\u884c\u5217\u5f0f\u4e0d\u4e3a\u96f6\uff09<\/span>\u65b9\u9635\u624d\u6709\u7279\u5f81\u503c\u5206\u89e3\uff0c\u800c\u4efb\u4f55 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u00d7<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u77e9\u9635\u90fd\u5b58\u5728SVD\u3002<\/span><\/section>\n

        <\/span><\/section>\n


        <\/h2>\n

        \n
        \n
        \n
        \n
        \n

        <\/section>\n

        <\/section>\n
        \n
        <\/svg><\/section>\n<\/section>\n<\/section>\n
        \n

        4. \u5947\u5f02\u503c\u5206\u89e3\u7684\u5e94\u7528<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

        \n

        <\/section>\n

        <\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


        <\/p>\n

        4.1 \u4f2a\u9006<\/span><\/strong><\/span><\/h3>\n

        <\/span><\/section>\n
        \u6211\u4eec\u53ef\u4ee5\u5229\u7528\u5947\u5f02\u503c\u5206\u89e3\u8ba1\u7b97\u77e9\u9635\u7684\u4f2a\u9006<\/span>\uff08\u77e9\u9635\u7684\u5e7f\u4e49\u9006\uff0c\u7528\u4e8e\u6c42\u89e3\u65b9\u7a0b\uff09<\/span>\u3002\u8bbe\u77e9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5947\u5f02\u503c\u5206\u89e3\u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>=<\/span><\/mo>U<\/span><\/mi><\/mrow>\u03a3<\/span><\/mi><\/mrow>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u5219\u5176\u4f2a\u9006\u4e3a\uff1a<\/span><\/section>\n


        <\/span><\/p>\n

        <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/p>\n

        <\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u8fd9\u91cc\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow>+<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u4f2a\u9006\uff0c\u6211\u4eec\u901a\u8fc7\u5c06\u6bcf\u4e2a\u975e\u96f6\u5bf9\u89d2\u5143\u7d20\u53d6\u5012\u6570\u5e76\u8f6c\u7f6e\u6765\u5f97\u5230\u5b83\u3002\u4f2a\u9006\u5728\u89e3\u51b3\u7ebf\u6027\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898<\/span>\uff08linear least squares problems\uff09<\/span>\u65f6\u5e38\u5e38\u6d3e\u4e0a\u7528\u573a\u3002<\/span><\/section>\n

        <\/span><\/section>\n

        4.2 \u6c42\u89e3\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b<\/span><\/strong><\/span><\/h3>\n

        <\/span><\/section>\n
        \u6211\u4eec\u53ef\u4ee5\u5c06\u4e00\u7ec4\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b<\/span>\uff08homogeneous linear equations\uff09<\/span>\u8868\u793a\u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow>x<\/span><\/mi><\/mrow>=<\/span><\/mo>0<\/span><\/mn><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u5176\u4e2d <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u77e9\u9635\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>x<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u5411\u91cf\u3002\u901a\u5e38\uff0c\u6211\u4eec\u5df2\u77e5 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u9700\u8981\u627e\u51fa\u6ee1\u8db3\u65b9\u7a0b\u7684\u975e\u96f6 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>x<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u8fd9\u6837\u7684 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>x<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5c5e\u4e8e <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u96f6\u7a7a\u95f4\uff0c\u4e5f\u79f0\u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684<\/span>\uff08\u53f3\uff09<\/span>\u96f6\u5411\u91cf\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u5411\u91cf <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>x<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u63cf\u8ff0\u4e3a\u4e0e <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u96f6\u5947\u5f02\u503c\u5bf9\u5e94\u7684\u53f3\u5947\u5f02\u5411\u91cf\u3002\u8fd9\u610f\u5473\u7740\uff0c\u5982\u679c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u65b9\u9635\u4e14\u6ca1\u6709\u96f6\u5947\u5f02\u503c\uff0c\u5219\u65b9\u7a0b\u6ca1\u6709\u975e\u96f6\u89e3\u3002\u82e5\u6709\u591a\u4e2a\u96f6\u5947\u5f02\u503c\uff0c\u90a3\u4e48\u5bf9\u5e94\u7684\u53f3\u5947\u5f02\u5411\u91cf\u7684\u4efb\u610f\u7ebf\u6027\u7ec4\u5408\u90fd\u662f\u6709\u6548\u89e3\u3002\u7c7b\u4f3c\u5730\uff0c\u6ee1\u8db3 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>x<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup>A<\/span><\/mi><\/mrow>=<\/span><\/mo>0<\/span><\/mn><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u975e\u96f6 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>x<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff08\u5176\u4e2d <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>x<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/span><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>x<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5171\u8f6d\u8f6c\u7f6e\uff09<\/span>\u88ab\u79f0\u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5de6\u96f6\u5411\u91cf\u3002<\/span><\/section>\n

        <\/span><\/section>\n

        4.3 \u603b\u4f53\u6700\u5c0f\u4e8c\u4e58\u6700\u5c0f\u5316<\/span><\/strong><\/span><\/h3>\n

        <\/span><\/section>\n
        \u603b\u4f53\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898<\/span>\uff08total least squares\uff09<\/span>\u65e8\u5728\u627e\u5230\u4e00\u4e2a\u5411\u91cf <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>x<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u4f7f\u5f97\u5728 <\/span><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u2016<\/span><\/mo>x<\/span><\/mi><\/mrow>\u2016<\/span><\/mo>=<\/span><\/mo>1<\/span><\/mn><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u7ea6\u675f\u4e0b\uff08\u7528\u53cc\u7ad6\u7ebf “||” \u5305\u56f4\u5411\u91cf\u8868\u793a\u5bf9\u8be5\u5411\u91cf\u6c42\u8303\u6570\uff09\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow>x<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u76842-\u8303\u6570\uff082-norm\uff09\u6700\u5c0f\u3002\u7ed3\u679c\u8868\u660e\uff0c\u89e3\u5c31\u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u6700\u5c0f\u5947\u5f02\u503c\u5bf9\u5e94\u7684\u53f3\u5947\u5f02\u5411\u91cf\u3002<\/span><\/section>\n

        <\/span><\/section>\n

        4.4 \u503c\u57df\u3001\u96f6\u7a7a\u95f4\u548c\u79e9<\/span><\/strong><\/span><\/h3>\n

        <\/span><\/section>\n
        SVD\u8fd8\u80fd\u4e3a\u77e9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u503c\u57df\u548c\u96f6\u7a7a\u95f4\u63d0\u4f9b\u660e\u786e\u8868\u793a\u3002<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u96f6\u5947\u5f02\u503c\u5bf9\u5e94\u7684\u53f3\u5947\u5f02\u5411\u91cf\u5f20\u6210<\/span>\uff08span\uff09<\/span>\u5176\u96f6\u7a7a\u95f4\uff0c\u975e\u96f6\u5947\u5f02\u503c\u5bf9\u5e94\u7684\u5de6\u5947\u5f02\u5411\u91cf\u5f20\u6210\u5176\u503c\u57df\u3002\u4f8b\u5982\uff0c\u5728\u524d\u9762\u7684\u4f8b\u5b50\u4e2d\uff0c\u96f6\u7a7a\u95f4\u7531 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u6700\u540e <\/span><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>(<\/span><\/mo>n<\/span><\/mi>\u2212<\/span><\/mo>r<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5217\u5f20\u6210\uff0c\u503c\u57df\u7531 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u524d <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>r<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5217\u5f20\u6210\uff0c\u5176\u4e2d <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>r<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u77e9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u79e9\u3002<\/span><\/section>\n


        <\/span><\/p>\n

        \u56e0\u6b64\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u79e9\u7b49\u4e8e\u975e\u96f6\u5947\u5f02\u503c\u7684\u4e2a\u6570\uff0c\u4e5f\u5c31\u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e2d\u975e\u96f6\u5bf9\u89d2\u5143\u7d20\u7684\u4e2a\u6570\u3002\u5728\u6570\u503c\u7ebf\u6027\u4ee3\u6570\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u5947\u5f02\u503c\u786e\u5b9a\u77e9\u9635\u7684\u6709\u6548\u79e9\uff0c\u56e0\u4e3a\u820d\u5165\u8bef\u5dee<\/span>\uff08rounding error\uff09<\/span>\u53ef\u80fd\u5bfc\u81f4\u79e9\u4e8f\u77e9\u9635<\/span>\uff08rank deficiency matrix\uff09<\/span>\u51fa\u73b0\u5c0f\u4f46\u975e\u96f6\u7684\u5947\u5f02\u503c\u3002\u6211\u4eec\u901a\u5e38\u8ba4\u4e3a\u8d85\u8fc7\u663e\u8457\u95f4\u9699\u7684\u5947\u5f02\u503c\u5728\u6570\u503c\u4e0a\u7b49\u540c\u4e8e\u96f6\u3002<\/span><\/section>\n

        4.5 \u4f4e\u79e9\u77e9\u9635\u8fd1\u4f3c<\/span><\/strong><\/span><\/h3>\n


        <\/span><\/p>\n

        \u4e00\u4e9b\u5b9e\u9645\u5e94\u7528\u9700\u8981\u7528\u53e6\u4e00\u4e2a\u7279\u5b9a\u79e9 r \u7684\u77e9\u9635 <\/span><\/sub><\/mo><\/mover><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff08\u79f0\u4e3a\u622a\u65ad\u77e9\u9635truncated\uff09<\/span>\u6765\u8fd1\u4f3c\u77e9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u5982\u679c\u6211\u4eec\u8981\u5728 <\/span><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-over><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-base><\/mjx-mover><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>rank<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>(<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/mi><\/sub><\/mo><\/mover><\/mrow>)<\/span><\/mo>=<\/span><\/mo>r<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u7ea6\u675f\u4e0b\u6700\u5c0f\u5316 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/sub><\/mo><\/mover><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e4b\u95f4\u5dee\u7684Frobenius\u8303\u6570\uff0c\u90a3\u4e48\u89e3\u5c31\u7531 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684SVD\u7ed9\u51fa\uff1a<\/span><\/section>\n
        <\/mjx-c><\/mjx-mo><\/mjx-over><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-base><\/mjx-mover><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-over><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-base><\/mjx-mover><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mover><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/p>\n

        <\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n
        \u8fd9\u91cc\uff0c<\/span><\/sub><\/span><\/mo><\/mover><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e0e <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u76f8\u540c\uff0c\u4f46\u53ea\u4fdd\u7559 r \u4e2a\u6700\u5927\u7684\u5947\u5f02\u503c<\/span>\uff08\u5176\u4ed6\u5947\u5f02\u503c\u7f6e\u96f6\uff09<\/span>\u3002\u8fd9\u5c31\u662fEckart\u2013Young\u5b9a\u7406\uff0c\u7531\u8fd9\u4e24\u4f4d\u4f5c\u8005\u4e8e1936\u5e74\u8bc1\u660e<\/span>\uff08\u5c3d\u7ba1\u540e\u6765\u53d1\u73b0\u65e9\u671f\u5b66\u8005\u5df2\u7ecf\u4e86\u89e3\u8fd9\u4e00\u7ed3\u679c\uff09<\/span>[1]\u3002<\/span><\/section>\n

        4.6 \u53ef\u5206\u79bb\u6a21\u578b<\/span><\/strong><\/span><\/h3>\n


        <\/span><\/p>\n

        \u6211\u4eec\u53ef\u4ee5\u5c06SVD\u89c6\u4e3a\u628a\u77e9\u9635\u5206\u89e3\u6210\u52a0\u6743\u3001\u6709\u5e8f\u7684\u53ef\u5206\u79bb\u77e9\u9635\u4e4b\u548c\u3002\u6240\u8c13\u53ef\u5206\u79bb\uff0c\u6307\u7684\u662f\u77e9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u53ef\u4ee5\u8868\u793a\u4e3a\u4e24\u4e2a\u5411\u91cf\u7684\u5916\u79ef<\/span>\uff08outer product \u4e24\u4e2a\u5411\u91cf\u7684\u53cd\u5bf9\u79f0\u79ef\uff0c\u751f\u6210\u77e9\u9635\uff09<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow>=<\/span><\/mo>u<\/span><\/mi><\/mrow>\u2297<\/span><\/mo>v<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u7528\u5750\u6807\u8868\u793a\u5373 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi>i<\/span><\/sub><\/mi>j<\/span><\/sub><\/mi><\/mrow><\/msub>=<\/span><\/mo>u<\/span><\/mi>i<\/span><\/sub><\/mi><\/msub>v<\/span><\/mi>j<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u77e9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5206\u89e3\u5982\u4e0b\uff1a<\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-base><\/mjx-row><\/mjx-c><\/mjx-mi><\/mjx-under><\/mjx-row><\/mjx-munder><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-base><\/mjx-row><\/mjx-c><\/mjx-mi><\/mjx-under><\/mjx-row><\/mjx-munder><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n
        <\/section>\n
        <\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n
        \u8fd9\u91cc\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>i<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>i<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5206\u522b\u662fSVD\u77e9\u9635\u7684\u7b2ci\u5217\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>i<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u6709\u5e8f\u5947\u5f02\u503c\uff0c\u6bcf\u4e2a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow>i<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u90fd\u662f\u53ef\u5206\u79bb\u7684\u3002\u5728\u56fe\u50cf\u5904\u7406\u4e2d\uff0c\u6211\u4eec\u5e38\u7528SVD\u5c06\u6ee4\u6ce2\u5668\u5206\u89e3\u4e3a\u6c34\u5e73\u548c\u5782\u76f4\u7684\u53ef\u5206\u79bb\u6ee4\u6ce2\u5668\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u975e\u96f6 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>i<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u6570\u91cf\u6070\u597d\u7b49\u4e8e\u77e9\u9635\u7684\u79e9\u3002<\/span><\/section>\n
        \u53ef\u5206\u79bb\u6a21\u578b\u5728\u751f\u7269\u7cfb\u7edf\u4e2d\u5f88\u5e38\u89c1\uff0cSVD\u5206\u89e3\u5728\u5206\u6790\u8fd9\u4e9b\u7cfb\u7edf\u65f6\u975e\u5e38\u6709\u7528\u3002\u4f8b\u5982\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u7a7a\u95f4\u57df\u7684Gabor\u6ee4\u6ce2\u5668\u4e58\u4ee5\u65f6\u95f4\u57df\u7684\u8c03\u5236\u51fd\u6570\u6765\u5f88\u597d\u5730\u63cf\u8ff0\u4e00\u4e9b\u89c6\u89c9\u533aV1\u7b80\u5355\u7ec6\u80de\u7684\u611f\u53d7\u91ce[2]\u3002\u56e0\u6b64\uff0c\u5982\u679c\u6211\u4eec\u901a\u8fc7\u53cd\u5411\u76f8\u5173<\/span>\uff08reverse correlation\uff09<\/span>\u7b49\u65b9\u6cd5\u8bc4\u4f30\u5f97\u5230\u7ebf\u6027\u6ee4\u6ce2\u5668\uff0c\u5c31\u53ef\u4ee5\u5c06\u4e24\u4e2a\u7a7a\u95f4\u7ef4\u5ea6\u91cd\u6392\u4e3a\u4e00\u4e2a\u7ef4\u5ea6\uff0c\u5f97\u5230\u4e00\u4e2a\u4e8c\u7ef4\u6ee4\u6ce2\u5668<\/span>\uff08\u7a7a\u95f4\uff0c\u65f6\u95f4\uff09<\/span>\uff0c\u7136\u540e\u8fdb\u884cSVD\u5206\u89e3\u3002\u5728SVD\u5206\u89e3\u4e2d\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u7b2c\u4e00\u5217\u5c31\u662fGabor\uff0c\u800c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u7b2c\u4e00\u5217\u4ee3\u8868\u65f6\u95f4\u8c03\u5236<\/span>\uff08\u6216\u53cd\u4e4b\uff09<\/span>\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\u4e00\u4e2a\u53ef\u5206\u79bb\u6027\u6307\u6570\uff1a<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-nstrut><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-spacer><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msubsup><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-munder><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-spacer><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msubsup><\/mjx-mrow><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n
        <\/section>\n
        <\/mn><\/msubsup><\/mrow><\/mfrac><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u8be5\u6307\u6570\u8868\u660e\u5728\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u5e42\u6b21\u4e2d\uff0c\u7b2c\u4e00\u4e2a\u53ef\u5206\u79bb\u77e9\u9635\u6240\u5360\u7684\u6bd4\u4f8b\u3002[3]\u3002<\/span><\/section>\n


        <\/span><\/h3>\n

        4.7 \u6700\u8fd1\u6b63\u4ea4\u77e9\u9635<\/span><\/strong><\/span><\/h3>\n

        <\/span><\/section>\n
        \u6211\u4eec\u53ef\u4ee5\u5229\u7528\u65b9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684SVD\u6765\u627e\u51fa\u6700\u63a5\u8fd1 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u6b63\u4ea4\u77e9\u9635<\/span>\uff08\u5217\u5411\u91cf\u4e92\u76f8\u6b63\u4ea4\u4e14\u5355\u4f4d\u957f\u5ea6\uff09 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>O<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u8fd9\u91cc\uff0c\u6211\u4eec\u7528 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>O<\/span><\/mi><\/mrow>\u2212<\/span><\/mo>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684Frobenius\u8303\u6570\u6765\u8861\u91cf\u63a5\u8fd1\u7a0b\u5ea6\u3002\u89e3\u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u4e58\u79ef[4]\u3002\u8fd9\u4e2a\u7ed3\u679c\u5728\u76f4\u89c9\u4e0a\u662f\u5408\u7406\u7684\uff0c\u56e0\u4e3a\u6b63\u4ea4\u77e9\u9635\u4f1a\u6709\u5206\u89e3 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>I<\/span><\/mi><\/mrow>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u5176\u4e2d <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>I<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u5355\u4f4d\u77e9\u9635\u3002\u6240\u4ee5\u5982\u679c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow>=<\/span><\/mo>U<\/span><\/mi><\/mrow>\u03a3<\/span><\/mi><\/mrow>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u90a3\u4e48\u4e58\u79ef <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow>=<\/span><\/mo>U<\/span><\/mi><\/mrow>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u76f8\u5f53\u4e8e\u75281\u66ff\u6362\u6240\u6709\u5947\u5f02\u503c\u3002\u7b49\u4ef7\u5730\uff0c\u89e3\u5c31\u662f\u6781\u5206\u89e3 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>=<\/span><\/mo>R<\/span><\/mi><\/mrow>P<\/span><\/mi><\/mrow>=<\/span><\/mo>P<\/span><\/mi><\/mrow>\u2032<\/span><\/mo><\/msup>R<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e2d\u7684\u9149\u77e9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>R<\/span><\/mi><\/mrow>=<\/span><\/mo>U<\/span><\/mi><\/mrow>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u65e0\u8bba\u62c9\u4f38\u548c\u65cb\u8f6c\u7684\u987a\u5e8f\u5982\u4f55\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u5728\u5f62\u72b6\u5206\u6790\u4e2d\uff0c\u6709\u4e00\u4e2a\u7c7b\u4f3c\u7684\u95ee\u9898\u53eb\u505a\u6b63\u4ea4\u666e\u9c81\u514b\u95ee\u9898<\/span>\uff08orthogonal Procrustes problem\uff09<\/span>\uff0c\u5b83\u6d89\u53ca\u627e\u5230\u4e00\u4e2a\u6700\u63a5\u8fd1\u5c06 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u6620\u5c04\u5230 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>B<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u6b63\u4ea4\u77e9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>O<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u5177\u4f53\u6765\u8bf4\uff1a<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mi><\/mjx-base><\/mjx-row><\/mjx-c><\/mjx-mi><\/mjx-under><\/mjx-row><\/mjx-munder><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-mspace><\/mjx-mstyle><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mtext><\/mjx-mspace><\/mjx-mstyle><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/p>\n

        <\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u5176\u4e2d <\/span><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u2016<\/span><\/mo>\u22c5<\/span><\/mo>\u2016<\/span><\/mo>F<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u8868\u793aFrobenius\u8303\u6570\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u8fd9\u4e2a\u95ee\u9898\u7b49\u540c\u4e8e\u4e3a\u7ed9\u5b9a\u77e9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>=<\/span><\/mo>A<\/span><\/mi><\/mrow>T<\/span><\/sup><\/mi><\/msup>B<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u627e\u5230\u6700\u8fd1\u7684\u6b63\u4ea4\u77e9\u9635\u3002<\/span><\/section>\n


        <\/span><\/h3>\n

        4.8 Kabsch\u7b97\u6cd5<\/span><\/strong><\/span><\/h3>\n

        <\/span><\/section>\n
        Kabsch\u7b97\u6cd5<\/span>\uff08\u5728\u5176\u4ed6\u9886\u57df\u79f0\u4e3aWahba\u95ee\u9898\uff09<\/span>\u8fd0\u7528SVD\u6765\u8ba1\u7b97\u6700\u4f18\u65cb\u8f6c<\/span>\uff08\u5173\u4e8e\u6700\u5c0f\u4e8c\u4e58\u6700\u5c0f\u5316\uff09<\/span>\uff0c\u4ee5\u5c06\u4e00\u7ec4\u70b9\u4e0e\u76f8\u5e94\u7684\u53e6\u4e00\u7ec4\u70b9\u5bf9\u9f50\u3002\u8fd9\u79cd\u7b97\u6cd5\u5728\u591a\u4e2a\u9886\u57df\u90fd\u6709\u5e94\u7528\uff0c\u6bd4\u5982\u7528\u4e8e\u6bd4\u8f83\u5206\u5b50\u7ed3\u6784\u3002<\/span><\/section>\n


        <\/span><\/h3>\n

        4.9 \u4fe1\u53f7\u5904\u7406<\/span><\/strong><\/span><\/h3>\n

        <\/span><\/section>\n
        \u7814\u7a76\u8005\u5df2\u6210\u529f\u5c06SVD\u548c\u4f2a\u9006\u5e94\u7528\u4e8e\u4fe1\u53f7\u5904\u7406 [5]\u3001\u56fe\u50cf\u5904\u7406 [6]\u548c\u5927\u6570\u636e<\/span>\uff08\u5982\u57fa\u56e0\u7ec4\u4fe1\u53f7\u5904\u7406\uff09<\/span>[7][8][9][10] \u3002<\/span><\/section>\n

        <\/span><\/section>\n

        4.10 \u5176\u4ed6\u4f8b\u5b50<\/span><\/strong><\/span><\/h3>\n

        <\/span><\/section>\n
        \u5947\u5f02\u503c\u5206\u89e3<\/span>(SVD)<\/span>\u5728\u7ebf\u6027\u53cd\u95ee\u9898<\/span>\uff08inverse problems\uff09<\/span>\u7814\u7a76\u4e2d\u5e7f\u6cdb\u5e94\u7528\uff0c\u5206\u6790Tikhonov\u6b63\u5219\u5316\u7b49\u65b9\u6cd5\u65f6\u9887\u6709\u52a9\u76ca\u3002\u7edf\u8ba1\u5b66\u754c\u666e\u904d\u4f7f\u7528\u5b83\uff0c\u4e0e\u4e3b\u6210\u5206\u5206\u6790<\/span>\uff08principal component analysis\uff09<\/span>\u548c\u5bf9\u5e94\u5206\u6790<\/span>\uff08correspondence analysis\uff09<\/span>\u5bc6\u5207\u76f8\u5173\uff0c\u4fe1\u53f7\u5904\u7406\u548c\u6a21\u5f0f\u8bc6\u522b\u9886\u57df\u4e5f\u5e38\u89c1\u5176\u8eab\u5f71\u3002\u6b64\u5916\uff0c\u5b83\u8fd8\u7528\u4e8e\u4ec5\u8f93\u51fa\u6a21\u6001\u5206\u6790<\/span>\uff08modal analysis\uff09<\/span>\uff0c\u53ef\u4ece\u5947\u5f02\u5411\u91cf\u786e\u5b9a\u975e\u7f29\u653e\u6a21\u6001\u5f62\u72b6<\/span>\uff08mode shapes\uff09<\/span>\u3002\u81ea\u7136\u8bed\u8a00\u6587\u672c\u5904\u7406\u4e2d\u7684\u6f5c\u5728\u8bed\u4e49\u7d22\u5f15<\/span>\uff08latent semantic indexing\uff09<\/span>\u4e5f\u79bb\u4e0d\u5f00\u5b83\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u5728\u6d89\u53ca\u7ebf\u6027\u6216\u7ebf\u6027\u5316\u7cfb\u7edf\u7684\u4e00\u822c\u6570\u503c\u8ba1\u7b97\u4e2d\uff0c\u5e38\u7528\u4e00\u4e2a\u666e\u904d\u5e38\u6570\u6765\u523b\u753b\u95ee\u9898\u7684\u89c4\u5f8b\u6027\u6216\u5947\u5f02\u6027\uff0c\u5373\u7cfb\u7edf\u7684”\u6761\u4ef6\u6570” <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mtext><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mtext><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03ba<\/span><\/mi>:=<\/span><\/mo>\u03c3<\/span><\/mi>max<\/span><\/sub><\/mtext><\/mrow><\/msub>\/<\/span><\/mo><\/mrow>\u03c3<\/span><\/mi>min<\/span><\/sub><\/mtext><\/mrow><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u8fd9\u4e2a\u6570\u503c\u901a\u5e38\u51b3\u5b9a\u4e86\u7ed9\u5b9a\u8ba1\u7b97\u65b9\u6848\u5728\u8fd9\u4e9b\u7cfb\u7edf\u4e0a\u7684\u8bef\u5dee\u7387\u6216\u6536\u655b\u901f\u5ea6[11][12]\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u91cf\u5b50\u4fe1\u606f<\/span>\uff08quantum information\uff09<\/span>\u9886\u57df\u4e2d\uff0cSVD\u4ee5Schmidt\u5206\u89e3\u7684\u5f62\u5f0f\u53d1\u6325\u7740\u5173\u952e\u4f5c\u7528\u3002\u901a\u8fc7\u5b83\uff0c\u6211\u4eec\u53ef\u4ee5\u81ea\u7136\u5730\u5206\u89e3\u4e24\u4e2a\u91cf\u5b50\u7cfb\u7edf\u7684\u72b6\u6001\uff0c\u4ece\u800c\u63d0\u4f9b\u4e86\u5b83\u4eec\u7ea0\u7f20\u7684\u5145\u8981\u6761\u4ef6\uff1a\u53ea\u8981 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u77e9\u9635\u7684\u79e9\u5927\u4e8e1\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u6570\u503c\u5929\u6c14\u9884\u62a5<\/span>\uff08numerical weather prediction\uff09<\/span>\u4e2d\uff0cSVD\u5bf9\u5927\u578b\u77e9\u9635\u4e5f\u6709\u91cd\u8981\u5e94\u7528\u3002\u5229\u7528Lanczos\u65b9\u6cd5\uff0c\u53ef\u4ee5\u4f30\u7b97\u5728\u7ed9\u5b9a\u521d\u59cb\u524d\u5411\u65f6\u95f4\u6bb5\u5185\uff0c\u5bf9\u4e2d\u5fc3\u6570\u503c\u5929\u6c14\u9884\u62a5\u7ebf\u6027\u589e\u957f\u6700\u5feb\u7684\u51e0\u4e2a\u6270\u52a8\u3002\u8fd9\u4e9b\u6270\u52a8\u5b9e\u9645\u4e0a\u662f\u8be5\u65f6\u95f4\u95f4\u9694\u5185\u5168\u7403\u5929\u6c14\u7ebf\u6027\u5316\u4f20\u64ad\u5b50\u5bf9\u5e94\u6700\u5927\u5947\u5f02\u503c\u7684\u5947\u5f02\u5411\u91cf\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u8f93\u51fa\u5947\u5f02\u5411\u91cf\u4ee3\u8868\u6574\u4e2a\u5929\u6c14\u7cfb\u7edf\u3002\u968f\u540e\uff0c\u8fd9\u4e9b\u6270\u52a8\u901a\u8fc7\u5b8c\u6574\u7684\u975e\u7ebf\u6027\u6a21\u578b\u8fd0\u884c\uff0c\u751f\u6210\u96c6\u5408\u9884\u62a5<\/span>\uff08ensemble forecast\uff09<\/span>\uff0c\u4e3a\u5f53\u524d\u4e2d\u5fc3\u9884\u6d4b\u5468\u56f4\u7684\u4e0d\u786e\u5b9a\u6027\u63d0\u4f9b\u4e86\u5904\u7406\u65b9\u6cd5\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u964d\u9636\u5efa\u6a21<\/span>\uff08reduced-order modeling\uff09<\/span>\u4e2d\u4e5f\u5c11\u4e0d\u4e86SVD\u7684\u8eab\u5f71\u3002\u964d\u9636\u5efa\u6a21\u65e8\u5728\u51cf\u5c11\u590d\u6742\u7cfb\u7edf\u4e2d\u7684\u81ea\u7531\u5ea6\u6570\u91cf\u3002\u7814\u7a76\u4eba\u5458\u5c06SVD\u4e0e\u5f84\u5411\u57fa\u51fd\u6570<\/span>\uff08radial basis functions\uff09<\/span>\u7ed3\u5408\uff0c\u7528\u4e8e\u63d2\u503c\u4e09\u7ef4\u975e\u7a33\u6001\u6d41\u95ee\u9898\u7684\u89e3[13]\u3002\u503c\u5f97\u4e00\u63d0\u7684\u662f\uff0c\u79d1\u5b66\u5bb6\u4eec\u5df2\u7ecf\u5229\u7528SVD\u6539\u8fdb\u4e86\u5730\u9762\u5f15\u529b\u6ce2\u5e72\u6d89\u4eeaaLIGO\u7684\u5f15\u529b\u6ce2\u5f62\u5efa\u6a21<\/span>\uff08gravitational wave modeling<\/span>\uff09[14]\u3002SVD\u6709\u52a9\u4e8e\u63d0\u9ad8\u6ce2\u5f62\u751f\u6210\u7684\u51c6\u786e\u6027\u548c\u901f\u5ea6\uff0c\u652f\u6301\u5f15\u529b\u6ce2\u641c\u7d22\u548c\u66f4\u65b0\u4e24\u79cd\u4e0d\u540c\u7684\u6ce2\u5f62\u6a21\u578b\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u63a8\u8350\u7cfb\u7edf<\/span>\uff08Recommender systems\uff09<\/span>\u4e2d\uff0cSVD\u7528\u4e8e\u9884\u6d4b\u7528\u6237\u5bf9\u9879\u76ee\u7684\u8bc4\u5206[15]\u3002\u4e3a\u4e86\u5728\u5546\u54c1\u673a\u5668\u96c6\u7fa4\u4e0a\u9ad8\u6548\u8ba1\u7b97SVD\uff0c\u7814\u7a76\u4eba\u5458\u5f00\u53d1\u4e86\u5206\u5e03\u5f0f\u7b97\u6cd5[16]\u3002<\/span><\/section>\n
        \u4f4e\u79e9SVD\u5728\u4ece\u65f6\u7a7a\u6570\u636e\u4e2d\u68c0\u6d4b\u70ed\u70b9\u65b9\u9762\u8868\u73b0\u51fa\u8272\uff0c\u5df2\u5e94\u7528\u4e8e\u75be\u75c5\u7206\u53d1\u68c0\u6d4b[17]\u3002\u7814\u7a76\u4eba\u5458\u8fd8\u5c06SVD\u548c\u9ad8\u9636SVD\u7ed3\u5408\u8d77\u6765\uff0c\u7528\u4e8e\u75be\u75c5\u76d1\u6d4b\u4e2d\u4ece\u590d\u6742\u6570\u636e\u6d41(\u5177\u6709\u7a7a\u95f4\u548c\u65f6\u95f4\u7ef4\u5ea6\u7684\u591a\u53d8\u91cf\u6570\u636e)\u8fdb\u884c\u5b9e\u65f6\u4e8b\u4ef6\u68c0\u6d4b[18]\u3002<\/span><\/section>\n
        \u5728\u5929\u4f53\u52a8\u529b\u5b66<\/span>\uff08astrodynamics\uff09<\/span>\u9886\u57df\uff0c\u79d1\u5b66\u5bb6\u4eec\u8fd0\u7528SVD\u53ca\u5176\u53d8\u4f53\u6765\u786e\u5b9a\u8f6c\u79fb\u8f68\u9053\u8bbe\u8ba1[19]\u548c\u8f68\u9053\u7ad9\u4fdd\u6301\u7684\u5408\u9002\u673a\u52a8\u65b9\u5411[20]\u3002<\/span><\/section>\n

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        <\/svg><\/section>\n<\/section>\n<\/section>\n
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        5. \u5b58\u5728\u6027\u8bc1\u660e<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

        \n

        <\/section>\n

        <\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


        <\/p>\n

        \u77e9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u7279\u5f81\u503c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03bb<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u6ee1\u8db3\u4ee3\u6570\u5173\u7cfb <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>u<\/span><\/mi><\/mrow>=<\/span><\/mo>\u03bb<\/span><\/mi>u<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u5f53 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u5384\u7c73\u77e9\u9635<\/span>\uff08Hermitian matrix \u81ea\u5171\u8f6d\u77e9\u9635\u7b49\u4e8e\u5176\u5171\u8f6d\u8f6c\u7f6e\uff09<\/span>\u65f6\uff0c\u6211\u4eec\u8fd8\u53ef\u4ee5\u7528\u53d8\u5206\u7279\u5f81\u6765\u63cf\u8ff0\u5b83\u3002\u5047\u8bbe <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u4e00\u4e2a <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>n<\/span><\/mi>\u00d7<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5b9e\u6570\u5bf9\u79f0\u77e9\u9635<\/span>\uff08\u77e9\u9635\u7b49\u4e8e\u5176\u8f6c\u7f6e\uff09<\/span>\uff0c\u6211\u4eec\u5b9a\u4e49\u51fd\u6570\uff1a<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-tstrut><\/mjx-mtd><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-tstrut><\/mjx-mtd><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-mo><\/mjx-mrow><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/p>\n

        <\/mi><\/mrow><\/mtd><\/mtr><\/mtable><\/mo><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u6781\u503c\u5b9a\u7406\u544a\u8bc9\u6211\u4eec\uff0c\u5f53\u8fd9\u4e2a\u8fde\u7eed\u51fd\u6570\u9650\u5236\u5728\u5355\u4f4d\u7403\u9762 <\/span><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>{<\/span><\/mo>\u2016<\/span><\/mo>x<\/span><\/mi><\/mrow>\u2016<\/span><\/mo>=<\/span><\/mo>1<\/span><\/mn>}<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e0a\u65f6\uff0c\u5fc5\u5728\u67d0\u70b9 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>u<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5904\u8fbe\u5230\u6700\u5927\u503c\u3002\u6839\u636e\u62c9\u683c\u6717\u65e5\u4e58\u6570\u5b9a\u7406\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>u<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5fc5\u7136\u6ee1\u8db3\uff1a<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/p>\n

        <\/mn><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u8fd9\u91cc <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03bb<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u67d0\u4e2a\u5b9e\u6570\uff0c\u800c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u2207<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u8868\u793a\u76f8\u5bf9\u4e8e <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>x<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684del\u5fae\u5206\u7b97\u5b50\u3002\u5229\u7528 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5bf9\u79f0\u6027\uff0c\u6211\u4eec\u5f97\u5230\uff1a<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/p>\n

        <\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u56e0\u6b64\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>u<\/span><\/mi><\/mrow>=<\/span><\/mo>\u03bb<\/span><\/mi>u<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u8fd9\u8bf4\u660e <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>u<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5355\u4f4d\u957f\u5ea6\u7279\u5f81\u5411\u91cf\u3002\u5bf9 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u6bcf\u4e2a\u5355\u4f4d\u957f\u5ea6\u7279\u5f81\u5411\u91cf <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>v<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u5176\u7279\u5f81\u503c\u5c31\u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>f<\/span><\/mi>(<\/span><\/mo>v<\/span><\/mi><\/mrow>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u6240\u4ee5 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03bb<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u6700\u5927\u7279\u5f81\u503c\u3002\u5728 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>u<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u6b63\u4ea4\u8865\u4e0a\u91cd\u590d\u8fd9\u4e2a\u8fc7\u7a0b\uff0c\u6211\u4eec\u5c31\u80fd\u5f97\u5230\u6b21\u5927\u7279\u5f81\u503c\uff0c\u4f9d\u6b64\u7c7b\u63a8\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u590d\u6570\u5384\u7c73\u77e9\u9635\u7684\u60c5\u51b5\uff0c\u5904\u7406\u65b9\u6cd5\u7c7b\u4f3c\uff0c\u53ea\u662f\u6b64\u65f6 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>f<\/span><\/mi>(<\/span><\/mo>x<\/span><\/mi><\/mrow>)<\/span><\/mo>=<\/span><\/mo>x<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup>M<\/span><\/mi><\/mrow>x<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u53d8\u6210\u4e86 2n \u4e2a\u5b9e\u53d8\u91cf\u7684\u5b9e\u503c\u51fd\u6570\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u5947\u5f02\u503c\u7684\u63cf\u8ff0\u65b9\u6cd5\u4e0e\u7279\u5f81\u503c\u76f8\u4f3c\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u4ee3\u6570\u6216\u53d8\u5206\u539f\u7406\u6765\u523b\u753b\u5b83\u4eec\u3002\u4e0d\u8fc7\uff0c\u4e0e\u7279\u5f81\u503c\u4e0d\u540c\uff0c\u8fd9\u91cc\u7684 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e0d\u5fc5\u662f\u5384\u7c73\u77e9\u9635\u6216\u5bf9\u79f0\u77e9\u9635\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u5c06\u4ece\u4e24\u4e2a\u89d2\u5ea6\u8bba\u8bc1\u5947\u5f02\u503c\u5206\u89e3\u7684\u5b58\u5728\u6027\u3002<\/span><\/section>\n


        <\/span><\/h3>\n

        5.1 \u57fa\u4e8e\u8c31\u5b9a\u7406<\/span><\/strong><\/span><\/h3>\n

        <\/span><\/section>\n
        \u8bbe <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u00d7<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u590d\u77e9\u9635\u3002\u7531\u4e8e <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u534a\u6b63\u5b9a\u5384\u7c73\u77e9\u9635\uff0c\u6839\u636e\u8c31\u5b9a\u7406\uff0c\u6211\u4eec\u53ef\u4ee5\u627e\u5230\u4e00\u4e2a <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>n<\/span><\/mi>\u00d7<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u9149\u77e9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u4f7f\u5f97\uff1a<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-over><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-base><\/mjx-mover><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/p>\n

        <\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u8fd9\u91cc <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>D<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u2113<\/span><\/mi>\u00d7<\/span><\/mo>\u2113<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7ef4\u7684\u5bf9\u89d2\u6b63\u5b9a\u77e9\u9635\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u2113<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u8868\u793a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u975e\u96f6\u7279\u5f81\u503c\u7684\u4e2a\u6570\uff08\u53ef\u4ee5\u8bc1\u660e <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u2113<\/span><\/mi>\u2264<\/span><\/mo>min<\/span><\/mo>(<\/span><\/mo>n<\/span><\/mi>,<\/span><\/mo>m<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff09\u3002\u6211\u4eec\u5c06 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5b9a\u4e49\u4e3a\u4e00\u4e2a\u77e9\u9635\uff0c\u5176\u7b2c i \u5217\u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5bf9\u5e94\u7279\u5f81\u503c <\/span><\/sub><\/mi><\/mrow><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u7279\u5f81\u5411\u91cf\u3002\u5f53 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>j<\/span><\/mi>><\/span><\/mo>\u2113<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u65f6\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u7b2c j \u5217\u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u96f6\u7279\u5f81\u503c\uff08\u5373<\/span><\/sub><\/mn><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff09\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u3002\u6211\u4eec\u53ef\u4ee5\u5c06 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5199\u6210 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-mspace><\/mjx-mstyle><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>=<\/span><\/mo>[<\/span><\/mo>V<\/span><\/mi><\/mrow>1 <\/span><\/sub><\/mn><\/msub><\/mspace><\/mstyle>V<\/span><\/mi><\/mrow>2<\/span><\/sub><\/mn><\/msub>]<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u5176\u4e2d <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>2<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5217\u5206\u522b\u5305\u542b <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u975e\u96f6\u548c\u96f6\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u3002\u7528\u8fd9\u79cd\u5f62\u5f0f\u91cd\u5199 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u65b9\u7a0b\u5c31\u53d8\u6210\uff1a<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-spacer><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msubsup><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-spacer><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msubsup><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mspace><\/mjx-mstyle><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-mspace><\/mjx-mstyle><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-spacer><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msubsup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-spacer><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msubsup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-spacer><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msubsup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-spacer><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msubsup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mo><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/p>\n

        <\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u8fd9\u610f\u5473\u7740\uff1a<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-spacer><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msubsup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-utext><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mjx-container><\/section>\n

        <\/p>\n

        <\/mo><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u8fdb\u4e00\u6b65\uff0c\u7b2c\u4e8c\u4e2a\u65b9\u7a0b\u8868\u660e <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>V<\/span><\/mi><\/mrow>2<\/span><\/sub><\/mn><\/msub>=<\/span><\/mo>0<\/span><\/mn><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u6b64\u5916\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u9149\u6027\u8d28\u5728 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>2<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u65b9\u9762\u4f53\u73b0\u4e3a\uff1a<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-spacer><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msubsup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-tstrut><\/mjx-mtd><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-utext><\/mjx-mi><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mjx-container><\/section>\n

        <\/p>\n

        <\/mi><\/mtd><\/mtr><\/mtable><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u8fd9\u91cc\u5355\u4f4d\u77e9\u9635\u7684\u4e0b\u6807\u7528\u6765\u8868\u793a\u5b83\u4eec\u7684\u4e0d\u540c\u7ef4\u5ea6\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u73b0\u5728\u6211\u4eec\u5b9a\u4e49\uff1a<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-nstrut><\/mjx-c><\/mjx-mn><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mn><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-utext><\/mjx-mo><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mjx-container><\/section>\n

        <\/p>\n

        <\/mo><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u90a3\u4e48\uff0c<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-nstrut><\/mjx-c><\/mjx-mn><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mn><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-spacer><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msubsup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-utext><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mjx-container><\/section>\n

        <\/p>\n


        <\/span><\/section>\n
        \u56e0\u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>V<\/span><\/mi><\/mrow>2<\/span><\/sub><\/mn><\/msub>=<\/span><\/mo>0<\/span><\/mn><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u8fd9\u4e5f\u53ef\u4ee5\u770b\u4f5c <\/span><\/sub><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u76f4\u63a5\u7ed3\u679c\u3002\u6362\u4e2a\u89d2\u5ea6\u770b\uff0c\u5982\u679c <\/span><\/sub><\/mi><\/msubsup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5bf9\u5e94\u975e\u96f6\u7279\u5f81\u503c <\/span><\/sub><\/mi><\/msubsup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u7279\u5f81\u5411\u91cf\u96c6\uff0c\u90a3\u4e48 <\/span><\/sub><\/mi><\/msubsup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5c31\u662f\u4e00\u7ec4\u6b63\u4ea4\u5411\u91cf\uff0c\u800c <\/span><\/sub><\/mi><\/msubsup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5219\u662f\u4e00\u7ec4<\/span>\uff08\u901a\u5e38\u4e0d\u5b8c\u5168\u7684\uff09<\/span>\u6b63\u4ea4\u5355\u4f4d\u5411\u91cf\u3002\u8fd9\u4e0e\u4e0a\u9762\u4f7f\u7528\u7684\u77e9\u9635\u5f62\u5f0f\u76f8\u7b26\uff0c\u5176\u4e2d <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u8868\u793a\u5217\u4e3a<\/span> <\/span><\/sub><\/mi><\/msubsup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> <\/span><\/mi><\/msubsup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u77e9\u9635\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>2<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u8868\u793a\u5217\u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u96f6\u7279\u5f81\u503c\u5bf9\u5e94\u7279\u5f81\u5411\u91cf\u7684\u77e9\u9635\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u8868\u793a\u5217\u4e3a\u5411\u91cf <\/span><\/sub><\/mi><\/msubsup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u77e9\u9635\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u6211\u4eec\u53d1\u73b0\u8fd9\u51e0\u4e4e\u5c31\u662f\u6240\u9700\u7684\u7ed3\u679c\uff0c\u53ea\u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u901a\u5e38\u4e0d\u662f\u9149\u77e9\u9635\uff0c\u56e0\u4e3a\u5b83\u4eec\u53ef\u80fd\u4e0d\u662f\u65b9\u9635\u3002\u7136\u800c\uff0c\u6211\u4eec\u77e5\u9053 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u884c\u6570\u4e0d\u5c0f\u4e8e\u5217\u6570\uff0c\u56e0\u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>D<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u7ef4\u5ea6\u4e0d\u5927\u4e8e m \u548c n\u3002\u6b64\u5916\uff0c\u7531\u4e8e<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-spacer><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msubsup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-nstrut><\/mjx-c><\/mjx-mn><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mn><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-spacer><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msubsup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-nstrut><\/mjx-c><\/mjx-mn><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mn><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-nstrut><\/mjx-c><\/mjx-mn><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mn><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-nstrut><\/mjx-c><\/mjx-mn><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mn><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-utext><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mjx-container><\/section>\n

        <\/p>\n

        <\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/mi><\/mrow><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e2d\u7684\u5217\u662f\u6b63\u4ea4\u5355\u4f4d\u7684\uff0c\u53ef\u4ee5\u6269\u5c55\u6210\u4e00\u4e2a\u6b63\u4ea4\u57fa\u3002\u8fd9\u610f\u5473\u7740\u6211\u4eec\u53ef\u4ee5\u9009\u62e9 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>2<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u4f7f\u5f97 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-mspace><\/mjx-mstyle><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>=<\/span><\/mo>[<\/span><\/mo>U<\/span><\/mi><\/mrow>1 <\/span><\/sub><\/mn><\/msub><\/mspace><\/mstyle>U<\/span><\/mi><\/mrow>2<\/span><\/sub><\/mn><\/msub>]<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u6210\u4e3a\u9149\u77e9\u9635\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u5bf9\u4e8e <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u6211\u4eec\u5df2\u7ecf\u6709 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>2<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4f7f\u5176\u6210\u4e3a\u9149\u77e9\u9635\u3002\u73b0\u5728\uff0c\u6211\u4eec\u5b9a\u4e49<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-beg><\/mjx-c><\/mjx-ext><\/mjx-c><\/mjx-end><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-nstrut><\/mjx-c><\/mjx-mn><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mn><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mn><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-c><\/mjx-beg><\/mjx-c><\/mjx-ext><\/mjx-c><\/mjx-end><\/mjx-mark><\/mjx-stretchy-v><\/mjx-mo><\/mjx-mrow><\/mjx-utext><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mjx-container><\/section>\n

        <\/p>\n

        <\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u5176\u4e2d\u6211\u4eec\u6dfb\u52a0\u6216\u5220\u9664\u989d\u5916\u7684\u96f6\u884c\uff0c\u4f7f\u96f6\u884c\u7684\u6570\u91cf\u7b49\u4e8e <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>2<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> <\/span><\/sub>\u7684\u5217\u6570\uff0c\u4ece\u800c\u4f7f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u6574\u4f53\u7ef4\u5ea6\u7b49\u4e8e <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u00d7<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u90a3\u4e48<\/span><\/section>\n

        <\/section>\n

        <\/p>\n

        <\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u8fd9\u5c31\u662f\u6211\u4eec\u8981\u8bc1\u660e\u7684\u7ed3\u679c\uff1a<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-utext><\/mjx-mo><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mjx-container><\/section>\n

        <\/p>\n

        <\/mo><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u8fd9\u4e2a\u8bba\u8bc1\u4e5f\u53ef\u4ee5\u4ece\u5bf9\u89d2\u5316 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u800c\u4e0d\u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5f00\u59cb<\/span>\uff08\u8fd9\u76f4\u63a5\u663e\u793a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/span><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/span><\/mo><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5177\u6709\u76f8\u540c\u7684\u975e\u96f6\u7279\u5f81\u503c\uff09<\/span>\u3002<\/span><\/section>\n

        <\/span><\/section>\n

        5.2 \u57fa\u4e8e\u53d8\u5206\u8868\u5f81\uff08variational characterization\uff09<\/span><\/strong><\/span><\/h3>\n

        <\/span><\/section>\n
        \u5947\u5f02\u503c\u53ef\u63cf\u8ff0\u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>u<\/span><\/mi><\/mrow>T<\/span><\/sup><\/mi><\/mrow><\/mrow><\/msup>M<\/span><\/mi><\/mrow>v<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u6700\u5927\u503c\uff0c\u8fd9\u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5728\u7279\u5b9a\u5b50\u7a7a\u95f4\u4e0a\u7684\u51fd\u6570\u3002\u5f53\u8fbe\u5230\u8fd9\u4e9b\u6700\u5927\u503c\u65f6\uff0c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u503c\u5c31\u662f\u5947\u5f02\u5411\u91cf\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u8bbe <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u00d7<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5b9e\u77e9\u9635\u3002\u5b9a\u4e49 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>S<\/span><\/mi>k<\/span><\/sup><\/mi>\u2212<\/span><\/sup><\/mo>1<\/span><\/sup><\/mn><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e3a <\/span><\/sub><\/mi><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e2d\u7684\u5355\u4f4d <\/span><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>(<\/span><\/mo>k<\/span><\/mi>\u2212<\/span><\/mo>1<\/span><\/mn>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>-\u7403\u9762\uff0c\u5e76\u4ee4 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>(<\/span><\/mo>u<\/span><\/mi><\/mrow>,<\/span><\/mo>v<\/span><\/mi><\/mrow>)<\/span><\/mo>=<\/span><\/mo>u<\/span><\/mi><\/mrow>T<\/span><\/sup><\/mi><\/mrow><\/msup>M<\/span><\/mi><\/mrow>v<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u5176\u4e2d <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>u<\/span><\/mi><\/mrow>\u2208<\/span><\/mo>S<\/span><\/mi>m<\/span><\/sup><\/mi>\u2212<\/span><\/sup><\/mo>1<\/span><\/sup><\/mn><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>v<\/span><\/mi><\/mrow>\u2208<\/span><\/mo>S<\/span><\/mi>n<\/span><\/sup><\/mi>\u2212<\/span><\/sup><\/mo>1<\/span><\/sup><\/mn><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u8003\u8651\u51fd\u6570 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5728 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>S<\/span><\/mi>m<\/span><\/sup><\/mi>\u2212<\/span><\/sup><\/mo>1<\/span><\/sup><\/mn><\/mrow><\/msup>\u00d7<\/span><\/mo>S<\/span><\/mi>n<\/span><\/sup><\/mi>\u2212<\/span><\/sup><\/mo>1<\/span><\/sup><\/mn><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e0a\u7684\u9650\u5236\u3002\u7531\u4e8e <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>S<\/span><\/mi>m<\/span><\/sup><\/mi>\u2212<\/span><\/sup><\/mo>1<\/span><\/sup><\/mn><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>S<\/span><\/mi>n<\/span><\/sup><\/mi>\u2212<\/span><\/sup><\/mo>1<\/span><\/sup><\/mn><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u90fd\u662f\u7d27\u96c6<\/span>\uff08compact sets\uff09<\/span>\uff0c\u5b83\u4eec\u7684\u4e58\u79ef\u4e5f\u662f\u7d27\u96c6\u3002\u53c8\u56e0 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u8fde\u7eed\uff0c\u5b83\u5fc5\u5728 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>S<\/span><\/mi>m<\/span><\/sup><\/mi>\u2212<\/span><\/sup><\/mo>1<\/span><\/sup><\/mn><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e2d\u7684\u67d0\u5411\u91cf <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>u<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>S<\/span><\/mi>n<\/span><\/mi>\u2212<\/span><\/mo>1<\/span><\/mn><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e2d\u7684\u67d0\u5411\u91cf <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>v<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5904\u8fbe\u5230\u6700\u5927\u503c\u3002\u6211\u4eec\u5c06\u8fd9\u4e2a\u6700\u5927\u503c\u8bb0\u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u76f8\u5e94\u7684\u5411\u91cf\u8bb0\u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>u<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> <\/span><\/sub>\u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>v<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u7531\u4e8e <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>(<\/span><\/mo>u<\/span><\/mi><\/mrow>,<\/span><\/mo>v<\/span><\/mi><\/mrow>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u6700\u5927\u503c\uff0c\u5b83\u5fc5\u4e3a\u975e\u8d1f\u3002\u82e5\u4e3a\u8d1f\uff0c\u6539\u53d8 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>u<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u6216 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>v<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u7b26\u53f7\u5c31\u80fd\u4f7f\u5176\u53d8\u4e3a\u6b63\u503c\uff0c\u4ece\u800c\u66f4\u5927\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u73b0\u5728\u6211\u4eec\u63d0\u51fa\u4ee5\u4e0b\u9648\u8ff0: <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>u<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>v<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> <\/span><\/sub>\u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5de6\u53f3\u5947\u5f02\u5411\u91cf\uff0c\u5bf9\u5e94\u7684\u5947\u5f02\u503c\u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u8bc1\u660e\u5982\u4e0b\uff1a\u7c7b\u4f3c\u4e8e\u7279\u5f81\u503c\u7684\u60c5\u51b5\uff0c\u6839\u636e\u5047\u8bbe\uff0c\u8fd9\u4e24\u4e2a\u5411\u91cf\u6ee1\u8db3\u62c9\u683c\u6717\u65e5\u4e58\u6570\u65b9\u7a0b:<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/p>\n

        <\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n
        \u7ecf\u8fc7\u4e00\u4e9b\u4ee3\u6570\u8fd0\u7b97\uff0c\u6211\u4eec\u5f97\u5230:<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-tstrut><\/mjx-mtd><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-tstrut><\/mjx-mtd><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/p>\n

        <\/mo><\/mtd><\/mtr><\/mtable><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u4ece\u5de6\u4fa7\u5c06\u7b2c\u4e00\u4e2a\u65b9\u7a0b\u4e58\u4ee5 <\/span><\/sub><\/mtext><\/mrow><\/msubsup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u7b2c\u4e8c\u4e2a\u65b9\u7a0b\u4e58\u4ee5 <\/span><\/sub><\/mtext><\/mrow><\/msubsup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u5e76\u8003\u8651\u5230 <\/span><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u2016<\/span><\/mo>u<\/span><\/mi><\/mrow>\u2016<\/span><\/mo>=<\/span><\/mo>\u2016<\/span><\/mo>v<\/span><\/mi><\/mrow>\u2016<\/span><\/mo>=<\/span><\/mo>1<\/span><\/mn><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u6211\u4eec\u5f97\u5230\uff1a<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>1<\/span><\/sub><\/mn><\/msub>=<\/span><\/mo>2<\/span><\/mn>\u03bb<\/span><\/mi>1<\/span><\/sub><\/mn><\/msub>=<\/span><\/mo>2<\/span><\/mn>\u03bb<\/span><\/mi>2<\/span><\/sub><\/mn><\/msub>.<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u5c06\u6b64\u4ee3\u5165\u4e0a\u9762\u7684\u4e00\u5bf9\u65b9\u7a0b\u4e2d\uff0c\u5f97:<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-tstrut><\/mjx-mtd><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-tstrut><\/mjx-mtd><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/mi><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/p>\n

        <\/mo><\/mtd><\/mtr><\/mtable><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/mo><\/mtd><\/mtr><\/mtable><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n
        \u8fd9\u5c31\u8bc1\u660e\u4e86\u8be5\u9648\u8ff0\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u8981\u627e\u5230\u66f4\u591a\u7684\u5947\u5f02\u5411\u91cf\u548c\u5947\u5f02\u503c\uff0c\u6211\u4eec\u53ef\u4ee5\u5728\u6b63\u4ea4\u4e8e <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>u<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>v<\/span><\/mi><\/mrow>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5f52\u4e00\u5316 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>u<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>v<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e0a\u6700\u5927\u5316 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>(<\/span><\/mo>u<\/span><\/mi><\/mrow>,<\/span><\/mo>v<\/span><\/mi><\/mrow>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u4ece\u5b9e\u6570\u5230\u590d\u6570\u7684\u8fc7\u6e21\u4e0e\u7279\u5f81\u503c\u7684\u60c5\u51b5\u7c7b\u4f3c\u3002<\/span><\/section>\n

        <\/span><\/section>\n


        <\/h2>\n

        \n
        \n
        \n
        \n
        \n

        <\/section>\n

        <\/section>\n
        \n
        <\/svg><\/section>\n<\/section>\n<\/section>\n
        \n

        6. \u8ba1\u7b97\u5947\u5f02\u503c\u5206\u89e3\uff08SVD\uff09<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

        \n

        <\/section>\n

        <\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


        <\/p>\n

        6.1 \u5355\u8fb9\u96c5\u53ef\u6bd4\u7b97\u6cd5\uff08One-sided Jacobi algorithm\uff09<\/span><\/strong><\/span><\/h3>\n

        <\/section>\n
        \u5355\u8fb9\u96c5\u53ef\u6bd4\u7b97\u6cd5\u662f\u4e00\u79cd\u8fed\u4ee3\u7b97\u6cd5[21]\uff0c\u5b83\u901a\u8fc7\u53cd\u590d\u8f6c\u6362\u4f7f\u77e9\u9635\u5217\u6b63\u4ea4\u5316\u3002\u5176\u57fa\u672c\u8fed\u4ee3\u6b65\u9aa4\u7531\u96c5\u53ef\u6bd4\u65cb\u8f6c<\/span>\uff08Jacobi rotation\uff09<\/span>\u7ed9\u51fa\uff1a<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi>\u2190<\/span><\/mo>M<\/span><\/mi>J<\/span><\/mi>(<\/span><\/mo>p<\/span><\/mi>,<\/span><\/mo>q<\/span><\/mi>,<\/span><\/mo>\u03b8<\/span><\/mi>)<\/span><\/mo>,<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u8fd9\u91cc\uff0c\u6211\u4eec\u9009\u62e9\u96c5\u53ef\u6bd4\u65cb\u8f6c\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>J<\/span><\/mi>(<\/span><\/mo>p<\/span><\/mi>,<\/span><\/mo>q<\/span><\/mi>,<\/span><\/mo>\u03b8<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u89d2\u5ea6<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03b8<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u4f7f\u65cb\u8f6c\u540e\u7b2c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>p<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5217\u548c\u7b2c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>q<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5217\u6b63\u4ea4\u3002\u6307\u6807<\/span><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>(<\/span><\/mo>p<\/span><\/mi>,<\/span><\/mo>q<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u6309<\/span><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>(<\/span><\/mo>p<\/span><\/mi>=<\/span><\/mo>1<\/span><\/mn>\u2026<\/span><\/mo>m<\/span><\/mi>,<\/span><\/mo>q<\/span><\/mi>=<\/span><\/mo>p<\/span><\/mi>+<\/span><\/mo>1<\/span><\/mn>\u2026<\/span><\/mo>m<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5faa\u73af\u626b\u63cf\uff0c\u5176\u4e2d<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e3a\u5217\u6570\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u7b97\u6cd5\u6536\u655b\u540e\uff0c\u6211\u4eec\u53ef\u4ee5\u8fd9\u6837\u6062\u590d\u5947\u5f02\u503c\u5206\u89e3<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi>=<\/span><\/mo>U<\/span><\/mi>S<\/span><\/mi>V<\/span><\/mi>T<\/span><\/sup><\/mi><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff1a\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u662f\u96c5\u53ef\u6bd4\u65cb\u8f6c\u77e9\u9635\u7684\u7d2f\u79ef\uff0c\u5c06\u8f6c\u6362\u540e\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u5217\u5f52\u4e00\u5316\u5f97\u5230\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u800c\u5947\u5f02\u503c\u5219\u7531\u8f6c\u6362\u540e\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5217\u7684\u8303\u6570\u7ed9\u51fa\u3002<\/span><\/section>\n


        <\/span><\/h3>\n

        6.2 \u53cc\u8fb9\u96c5\u53ef\u6bd4\u7b97\u6cd5\uff08Two-sided Jacobi algorithm\uff09<\/span><\/strong><\/span><\/h3>\n

        <\/span><\/section>\n
        \u53cc\u8fb9\u96c5\u53ef\u6bd4SVD\u7b97\u6cd5\u662f\u96c5\u53ef\u6bd4\u7279\u5f81\u503c\u7b97\u6cd5<\/span>\uff08Jacobi eigenvalue algorithm\uff09<\/span>\u7684\u63a8\u5e7f\uff0c\u5b83\u901a\u8fc7\u8fed\u4ee3\u5c06\u65b9\u9635\u8f6c\u6362\u4e3a\u5bf9\u89d2\u77e9\u9635\u3002\u5bf9\u4e8e\u975e\u65b9\u9635\uff0c\u6211\u4eec\u5148\u8fdb\u884cQR\u5206\u89e3\uff0c\u7136\u540e\u5bf9<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>R<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u77e9\u9635\u5e94\u7528\u6b64\u7b97\u6cd5\u3002\u5176\u57fa\u672c\u8fed\u4ee3\u6b65\u9aa4\u5982\u4e0b\uff1a\u5148\u7528Givens\u65cb\u8f6c\u4f7f\u4e00\u5bf9\u975e\u5bf9\u89d2\u5143\u7d20\u5bf9\u79f0\uff0c\u518d\u7528\u96c5\u53ef\u6bd4\u53d8\u6362<\/span>\uff08Jacobi transformation\uff09<\/span>\u4f7f\u5b83\u4eec\u5f52\u96f6\uff0c<\/span><\/section>\n

        <\/span><\/section>\n
        <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi>\u2190<\/span><\/mo>J<\/span><\/mi>T<\/span><\/sup><\/mi><\/mrow><\/msup>G<\/span><\/mi>M<\/span><\/mi>J<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

        <\/span><\/section>\n
        \u8fd9\u91cc\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>G<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u662fGivens\u65cb\u8f6c\u77e9\u9635\uff0c\u6211\u4eec\u9009\u62e9\u9002\u5f53\u7684\u89d2\u5ea6\u4f7f\u6307\u5b9a\u7684\u975e\u5bf9\u89d2\u5143\u7d20\u5bf9\u5728\u65cb\u8f6c\u540e\u76f8\u7b49\uff1b<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>J<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u662f\u96c5\u53ef\u6bd4\u53d8\u6362\u77e9\u9635\uff0c\u7528\u4e8e\u5c06\u8fd9\u4e9b\u975e\u5bf9\u89d2\u5143\u7d20\u7f6e\u96f6\u3002\u8fed\u4ee3\u8fc7\u7a0b\u4e0e\u96c5\u53ef\u6bd4\u7279\u5f81\u503c\u7b97\u6cd5\u76f8\u540c\uff1a\u5bf9\u6240\u6709\u975e\u5bf9\u89d2\u5143\u7d20\u5faa\u73af\u626b\u63cf\u3002<\/span><\/section>\n

        <\/span><\/section>\n
        \u7b97\u6cd5\u6536\u655b\u540e\uff0c\u7ed3\u679c\u5bf9\u89d2\u77e9\u9635\u5305\u542b\u5947\u5f02\u503c\u3002\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u7d2f\u79ef\u5982\u4e0b\uff1a<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi>\u2190<\/span><\/mo>U<\/span><\/mi>G<\/span><\/mi>T<\/span><\/sup><\/mi><\/mrow><\/msup>J<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi>\u2190<\/span><\/mo>V<\/span><\/mi>J<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002<\/span><\/section>\n


        <\/span><\/h3>\n

        6.3 \u6570\u503c\u65b9\u6cd5<\/span><\/strong><\/span><\/h3>\n

        <\/span><\/section>\n

        \u6211\u4eec\u53ef\u4ee5\u5229\u7528\u4ee5\u4e0b\u89c2\u5bdf\u7ed3\u679c\u8ba1\u7b97\u5947\u5f02\u503c\u5206\u89e3\uff1a<\/span><\/p>\n

          \n
        • \n
          <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u5de6\u5947\u5f02\u5411\u91cf\u662f<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u4e00\u7ec4\u6b63\u4ea4\u5f52\u4e00\u5316\u7279\u5f81\u5411\u91cf\u3002<\/span><\/section>\n<\/li>\n
        • \n
          <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u53f3\u5947\u5f02\u5411\u91cf\u662f<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/mrow><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u4e00\u7ec4\u6b63\u4ea4\u5f52\u4e00\u5316\u7279\u5f81\u5411\u91cf\u3002<\/span><\/section>\n<\/li>\n
        • \n

          <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u975e\u96f6\u5947\u5f02\u503c<\/span>\uff08\u5728<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u5bf9\u89d2\u7ebf\u4e0a\uff09<\/span>\u7b49\u4e8e<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/mrow><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u975e\u96f6\u7279\u5f81\u503c\u7684\u5e73\u65b9\u6839\u3002<\/span><\/p>\n<\/li>\n<\/ul>\n


          <\/span><\/section>\n
          \u901a\u5e38\uff0c\u6211\u4eec\u901a\u8fc7\u4e24\u6b65\u8ba1\u7b97\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684SVD\u3002\u7b2c\u4e00\u6b65\u5c06\u77e9\u9635\u7b80\u5316\u4e3a\u53cc\u5bf9\u89d2\u77e9\u9635<\/span>\uff08bidiagonal matrix\uff09<\/span>\uff0c\u5047\u8bbe<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u2265<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u9700\u8981<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>O<\/span><\/mi>(<\/span><\/mo>m<\/span><\/mi>n<\/span><\/mi>2<\/span><\/sup><\/mn><\/mrow><\/msup>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u6b21\u6d6e\u70b9\u8fd0\u7b97<\/span>\uff08flop\uff09<\/span>\u3002\u7b2c\u4e8c\u6b65\u8ba1\u7b97\u53cc\u5bf9\u89d2\u77e9\u9635\u7684SVD\uff0c\u4e14\u53ea\u80fd\u901a\u8fc7\u8fed\u4ee3\u65b9\u6cd5\u5b8c\u6210\u3002\u5b9e\u9645\u4e0a\uff0c\u8ba1\u7b97\u5230\u673a\u5668\u7cbe\u5ea6\u5c31\u8db3\u591f\u4e86\u3002\u5982\u679c\u5c06\u8fd9\u4e2a\u7cbe\u5ea6\u89c6\u4e3a\u5e38\u6570\uff0c\u7b2c\u4e8c\u6b65\u9700\u8981<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>O<\/span><\/mi>(<\/span><\/mo>n<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u6b21\u8fed\u4ee3\uff0c\u6bcf\u6b21\u8fed\u4ee3\u82b1\u8d39<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>O<\/span><\/mi>(<\/span><\/mo>n<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u6b21flop\u3002\u56e0\u6b64\uff0c\u7b2c\u4e00\u6b65\u8017\u65f6\u66f4\u957f\uff0c\u603b\u4f53\u6210\u672c\u4e3a<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>O<\/span><\/mi>(<\/span><\/mo>m<\/span><\/mi>n<\/span><\/mi>2<\/span><\/sup><\/mn><\/mrow><\/msup>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u6b21flop[22]\u3002<\/span><\/section>\n

          <\/span><\/section>\n
          \u5982\u679c\u53ea\u9700\u8ba1\u7b97\u5947\u5f02\u503c\uff0c\u7b2c\u4e00\u6b65\u53ef\u7528Householder\u53cd\u5c04\u4ee5<\/span><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>4<\/span><\/mn>m<\/span><\/mi>n<\/span><\/mi>2<\/span><\/sup><\/mn><\/mrow><\/msup>\u2212<\/span><\/mo>4<\/span><\/mn>n<\/span><\/mi>3<\/span><\/sup><\/mn><\/mrow><\/msup>\/<\/span><\/mo><\/mrow>3<\/span><\/mn><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u6b21flop\u5b8c\u6210\u3002\u5f53<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u8fdc\u5927\u4e8e<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u65f6\uff0c\u5148\u7528QR\u5206\u89e3\u5c06\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7b80\u5316\u4e3a\u4e09\u89d2\u77e9\u9635\uff0c\u518d\u7528Householder\u53cd\u5c04\u8fdb\u4e00\u6b65\u7b80\u5316\u4e3a\u53cc\u5bf9\u89d2\u5f62\u5f0f\u66f4\u6709\u4f18\u52bf\uff1b\u603b\u6210\u672c\u4e3a<\/span><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>2<\/span><\/mn>m<\/span><\/mi>n<\/span><\/mi>2<\/span><\/sup><\/mn><\/mrow><\/msup>+<\/span><\/mo>2<\/span><\/mn>n<\/span><\/mi>3<\/span><\/sup><\/mn><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u6b21flop[22]\u3002<\/span><\/section>\n

          <\/span><\/section>\n
          \u7b2c\u4e8c\u6b65\u53ef\u7528QR\u7b97\u6cd5\u7684\u53d8\u4f53\u5b8c\u6210\uff0c\u8be5\u53d8\u4f53\u7531Golub & Kahan[23]\u9996\u6b21\u63cf\u8ff0\u3002LAPACK\u5b50\u7a0b\u5e8fDBDSQR[24]\u5b9e\u73b0\u4e86\u8fd9\u79cd\u8fed\u4ee3\u65b9\u6cd5\uff0c\u5e76\u9488\u5bf9\u5947\u5f02\u503c\u975e\u5e38\u5c0f\u7684\u60c5\u51b5\u8fdb\u884c\u4e86\u6539\u8fdb[25]\u3002\u7ed3\u5408\u4f7f\u7528Householder\u53cd\u5c04\u7684\u7b2c\u4e00\u6b65\u548c\u9002\u5f53\u60c5\u51b5\u4e0b\u7684QR\u5206\u89e3\uff0c\u6784\u6210\u4e86\u8ba1\u7b97\u5947\u5f02\u503c\u5206\u89e3\u7684DGESVD[24]\u4f8b\u7a0b\u3002<\/span><\/section>\n

          <\/span><\/section>\n
          GNU\u79d1\u5b66\u5e93<\/span>\uff08GSL\uff09<\/span>\u4e5f\u5b9e\u73b0\u4e86\u76f8\u540c\u7b97\u6cd5\uff0c\u5e76\u63d0\u4f9b\u4e86\u4e00\u79cd\u66ff\u4ee3\u65b9\u6cd5\uff0c\u5728\u7b2c2\u6b65\u4e2d\u4f7f\u7528\u5355\u8fb9\u96c5\u53ef\u6bd4\u6b63\u4ea4\u5316[26]\u3002\u8fd9\u79cd\u65b9\u6cd5\u901a\u8fc7\u6c42\u89e3\u4e00\u7cfb\u5217<\/span><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>2<\/span><\/mn>\u00d7<\/span><\/mo>2<\/span><\/mn><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684SVD\u95ee\u9898\u6765\u8ba1\u7b97\u53cc\u5bf9\u89d2\u77e9\u9635\u7684SVD\uff0c\u7c7b\u4f3c\u4e8e\u96c5\u53ef\u6bd4\u7279\u5f81\u503c\u7b97\u6cd5\u6c42\u89e3\u4e00\u7cfb\u5217<\/span><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>2<\/span><\/mn>\u00d7<\/span><\/mo>2<\/span><\/mn><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u7279\u5f81\u503c\u95ee\u9898[27]\u3002\u7b2c2\u6b65\u7684\u53e6\u4e00\u79cd\u65b9\u6cd5\u501f\u9274\u4e86\u5206\u6cbb\u7279\u5f81\u503c\u7b97\u6cd5<\/span>\uff08divide-and-conquer eigenvalue algorithms\uff09<\/span>\u7684\u601d\u60f3[22]\u3002<\/span><\/section>\n

          <\/span><\/section>\n
          \u8fd8\u6709\u4e00\u79cd\u4e0d\u663e\u5f0f\u4f7f\u7528\u7279\u5f81\u503c\u5206\u89e3\u7684\u66ff\u4ee3\u65b9\u6cd5[28]\u3002\u901a\u5e38\uff0c\u6211\u4eec\u5c06\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u5947\u5f02\u503c\u95ee\u9898\u8f6c\u6362\u4e3a\u7b49\u4ef7\u7684\u5bf9\u79f0\u7279\u5f81\u503c\u95ee\u9898\uff0c\u5982<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/mrow><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u6216 <\/span><\/sub><\/span><\/mn><\/mtd><\/mtr><\/mtable><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002<\/span><\/span><\/section>\n

          <\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n
          \u4f7f\u7528\u7279\u5f81\u503c\u5206\u89e3\u7684\u65b9\u6cd5\u57fa\u4e8eQR\u7b97\u6cd5\uff0c\u8be5\u7b97\u6cd5\u5df2\u53d1\u5c55\u5f97\u7a33\u5b9a\u4e14\u5feb\u901f\u3002\u6ce8\u610f\uff0c\u5947\u5f02\u503c\u662f\u5b9e\u6570\uff0c\u53f3\u5947\u5f02\u5411\u91cf\u548c\u5de6\u5947\u5f02\u5411\u91cf\u4e0d\u9700\u8981\u5f62\u6210\u76f8\u4f3c\u53d8\u6362\u3002\u6211\u4eec\u53ef\u4ee5\u5728QR\u5206\u89e3\u548cLQ\u5206\u89e3\u4e4b\u95f4\u8fed\u4ee3\u4ea4\u66ff\u4ee5\u627e\u5230\u5b9e\u5bf9\u89d2\u5384\u7c73\u77e9\u9635\u3002QR\u5206\u89e3\u7ed9\u51fa<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u21d2<\/span><\/mo>Q<\/span><\/mi><\/mrow>R<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>R<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684LQ\u5206\u89e3\u7ed9\u51fa<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>R<\/span><\/mi><\/mrow>\u21d2<\/span><\/mo>L<\/span><\/mi><\/mrow>P<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u56e0\u6b64\uff0c\u6bcf\u6b21\u8fed\u4ee3\u4e2d\uff0c\u6211\u4eec\u6709<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u21d2<\/span><\/mo>Q<\/span><\/mi><\/mrow>L<\/span><\/mi><\/mrow>P<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u66f4\u65b0<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u21d0<\/span><\/mo>L<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5e76\u91cd\u590d\u6b63\u4ea4\u5316\u3002\u6700\u7ec8\uff0cQR\u5206\u89e3\u548cLQ\u5206\u89e3\u4e4b\u95f4\u7684\u8fd9\u79cd\u8fed\u4ee3\u4ea7\u751f\u5de6\u53f3\u9149\u5947\u5f02\u77e9\u9635\u3002\u8fd9\u79cd\u65b9\u6cd5\u4e0d\u80fd\u50cfQR\u7b97\u6cd5\u90a3\u6837\u901a\u8fc7\u8c31\u79fb\u4f4d\u6216\u7f29\u51cf\u6765\u52a0\u901f\uff0c\u56e0\u4e3a\u6ca1\u6709\u4f7f\u7528\u76f8\u4f3c\u53d8\u6362\uff0c\u79fb\u4f4d\u65b9\u6cd5\u4e0d\u6613\u5b9a\u4e49\u3002\u7136\u800c\uff0c\u8fd9\u79cd\u8fed\u4ee3\u65b9\u6cd5\u5b9e\u73b0\u7b80\u5355\uff0c\u5f53\u901f\u5ea6\u4e0d\u91cd\u8981\u65f6\u662f\u4e2a\u4e0d\u9519\u7684\u9009\u62e9\u3002\u5b83\u8fd8\u63d0\u4f9b\u4e86\u7eaf\u6b63\u4ea4\/\u9149\u53d8\u6362\u5982\u4f55\u83b7\u5f97SVD\u7684\u6d1e\u5bdf\u3002<\/span><\/section>\n

          <\/span><\/section>\n

          6.4 2 \u00d7 2 SVD\u7684\u89e3\u6790\u7ed3\u679c<\/span><\/strong><\/span><\/h3>\n

          <\/span><\/section>\n
          \u89e3\u6790<\/span><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>2<\/span><\/mn>\u00d7<\/span><\/mo>2<\/span><\/mn><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u6211\u4eec\u53ef\u4ee5\u627e\u5230\u77e9\u9635\u7684\u5947\u5f02\u503c\u3002\u8bbe\u77e9\u9635\u4e3a<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>=<\/span><\/mo>z<\/span><\/mi>0<\/span><\/sub><\/mn><\/mrow><\/msub>I<\/span><\/mi><\/mrow>+<\/span><\/mo>z<\/span><\/mi>1<\/span><\/sub><\/mn><\/mrow><\/msub>\u03c3<\/span><\/mi>1<\/span><\/sub><\/mn><\/mrow><\/msub>+<\/span><\/mo>z<\/span><\/mi>2<\/span><\/sub><\/mn><\/mrow><\/msub>\u03c3<\/span><\/mi>2<\/span><\/sub><\/mn><\/mrow><\/msub>+<\/span><\/mo>z<\/span><\/mi>3<\/span><\/sub><\/mn><\/mrow><\/msub>\u03c3<\/span><\/mi>3<\/span><\/sub><\/mn><\/mrow><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/span><\/sub><\/mn><\/mrow><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n
          \u5176\u4e2d<\/span> <\/span> <\/sub><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u662f\u53c2\u6570\u5316\u77e9\u9635\u7684\u590d\u6570\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>I<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u662f\u5355\u4f4d\u77e9\u9635\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>i<\/span><\/sub><\/mi><\/mrow><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u8868\u793a\u6ce1\u5229\u77e9\u9635<\/span>\uff08Pauli matrices\uff09<\/span>\u3002\u90a3\u4e48\u5b83\u7684\u4e24\u4e2a\u5947\u5f02\u503c\u7531\u4ee5\u4e0b\u7ed9\u51fa\uff1a<\/span><\/section>\n

          <\/span><\/section>\n
          <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-tstrut><\/mjx-mtd><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-surd><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-surd><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-spacer><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msubsup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-tstrut><\/mjx-mtd><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-surd><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-surd><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-spacer><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msubsup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/span><\/mi><\/msub><\/mtd><\/mtr><\/mtable><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/p>\n

          <\/span><\/mn><\/mrow><\/msup><\/msqrt><\/msqrt><\/mtd><\/mtr><\/mtable><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n


          <\/span><\/h2>\n


          <\/h2>\n

          \n
          \n
          \n
          \n
          \n

          <\/section>\n

          <\/section>\n
          \n
          <\/svg><\/section>\n<\/section>\n<\/section>\n
          \n

          7. \u7b80\u5316\u5947\u5f02\u503c\u5206\u89e3<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

          \n

          <\/section>\n

          <\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


          <\/p>\n

          <\/section>\n

          <\/span><\/section>\n
          \u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u5f88\u5c11\u9700\u8981\u5b8c\u5168SVD\uff0c\u5305\u62ec\u77e9\u9635\u96f6\u7a7a\u95f4\u7684\u5b8c\u6574\u9149\u5206\u89e3\u3002\u76f8\u53cd\uff0c\u4e0a\u56fe\u4e2d\u7684\u8ba1\u7b97\u7b80\u5316\u7248\u7684SVD\u901a\u5e38\u5c31\u591f\u7528\u4e86 <\/span>(\u800c\u4e14\u66f4\u5feb\uff0c\u66f4\u7701\u5b58\u50a8\u7a7a\u95f4)<\/span>\u3002\u5bf9\u4e8e\u4e00\u4e2a\u79e9\u4e3ar\u7684<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u00d7<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u6211\u4eec\u53ef\u4ee5\u533a\u5206\u4ee5\u4e0b\u51e0\u79cd\u60c5\u51b5:<\/span><\/section>\n

          <\/span><\/section>\n

          7.1 \u8584SVD (Thin SVD)<\/span><\/strong><\/span><\/h3>\n

          <\/span><\/section>\n
          \u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u8584SVD<\/span> (\u53c8\u53eb\u7ecf\u6d4e\u578bSVD)<\/span> \u53ef\u8868\u793a\u4e3a[29]:<\/span><\/section>\n

          <\/span><\/section>\n
          \u8fd9\u91cc<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>k<\/span><\/mi>=<\/span><\/mo>min<\/span><\/mo>(<\/span><\/mo>m<\/span><\/mi>,<\/span><\/mo>n<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>k<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>k<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u53ea\u5305\u542b<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u524dk\u5217\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow>k<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u53ea\u5305\u542b<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e2d\u7684\u524dk\u4e2a\u5947\u5f02\u503c\u3002\u56e0\u6b64\uff0c\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>k<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e3a<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u00d7<\/span><\/mo>k<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow>k<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e3a<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>k<\/span><\/mi>\u00d7<\/span><\/mo>k<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5bf9\u89d2\u77e9\u9635\uff0c<\/span><\/sub><\/span><\/mo><\/msubsup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e3a<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>k<\/span><\/mi>\u00d7<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002<\/span><\/section>\n

          <\/span><\/section>\n
          \u5f53<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>k<\/span><\/mi>\u226a<\/span><\/mo>max<\/span><\/mo>(<\/span><\/mo>m<\/span><\/mi>,<\/span><\/mo>n<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u65f6\uff0c\u8584SVD\u80fd\u5927\u5927\u8282\u7701\u7a7a\u95f4\u548c\u8ba1\u7b97\u65f6\u95f4\u3002\u5b83\u7684\u8ba1\u7b97\u901a\u5e38\u5148\u5bf9<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u505aQR\u5206\u89e3\uff0c\u8fd9\u6837\u80fd\u663e\u8457\u52a0\u5feb\u8fd0\u7b97\u901f\u5ea6\u3002<\/span><\/section>\n

          <\/span><\/section>\n

          7.2 \u7d27\u51d1SVD (Compact SVD)<\/span><\/strong><\/span><\/h3>\n

          <\/span><\/section>\n
          \u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u7d27\u51d1SVD\u53ef\u8868\u793a\u4e3a\uff1a<\/span><\/section>\n

          <\/span><\/section>\n
          <\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/p>\n

          <\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/span><\/section>\n
          \u6211\u4eec\u53ea\u8ba1\u7b97<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684r\u4e2a\u5217\u5411\u91cf\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684r\u4e2a\u884c\u5411\u91cf\uff0c\u5b83\u4eec\u5bf9\u5e94\u975e\u96f6\u5947\u5f02\u503c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow>r<\/span><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u5176\u4f59\u5411\u91cf\u5219\u4e0d\u8ba1\u7b97\u3002\u5f53<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>r<\/span><\/mi>\u226a<\/span><\/mo>min<\/span><\/mo>(<\/span><\/mo>m<\/span><\/mi>,<\/span><\/mo>n<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u65f6\uff0c\u8fd9\u6bd4\u8584SVD\u66f4\u5feb\u66f4\u7701\u3002\u56e0\u6b64\uff0c\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>r<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e3a<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u00d7<\/span><\/mo>r<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow>r<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e3a<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>r<\/span><\/mi>\u00d7<\/span><\/mo>r<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5bf9\u89d2\u77e9\u9635\uff0c<\/span><\/sub><\/span><\/mo><\/msubsup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e3a<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>r<\/span><\/mi>\u00d7<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002<\/span><\/section>\n

          <\/span><\/section>\n

          7.3 \u622a\u65adSVD (Truncated SVD)<\/span><\/strong><\/span><\/h3>\n

          <\/span><\/section>\n
          \u5728\u5f88\u591a\u5e94\u7528\u4e2d\uff0c\u975e\u96f6\u5947\u5f02\u503c\u7684\u6570\u91cfr\u5f88\u5927\uff0c\u4f7f\u5f97\u5373\u4f7f\u7d27\u51d1SVD\u4e5f\u96be\u4ee5\u8ba1\u7b97\u3002\u8fd9\u65f6\u6211\u4eec\u53ef\u80fd\u9700\u8981\u622a\u65ad\u6700\u5c0f\u7684\u5947\u5f02\u503c\uff0c\u53ea\u8ba1\u7b97<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>t<\/span><\/mi>\u226a<\/span><\/mo>r<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e2a\u975e\u96f6\u5947\u5f02\u503c\u3002\u622a\u65adSVD\u4e0d\u518d\u662f\u539f\u59cb\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u7cbe\u786e\u5206\u89e3\uff0c\u800c\u662f\u63d0\u4f9b\u4e86\u4e00\u4e2a\u56fa\u5b9a\u79e9t\u7684\u6700\u4f18\u4f4e\u79e9\u77e9\u9635\u8fd1\u4f3c<\/span>\uff08low-rank matrix approximation\uff09<\/span> <\/sub>:<\/span><\/section>\n

          <\/span><\/section>\n
          <\/mjx-c><\/mjx-mo><\/mjx-over><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-base><\/mjx-mover><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-spacer><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msubsup><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/span><\/mi><\/mrow><\/mover><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n
          <\/section>\n
          <\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/span><\/section>\n
          \u8fd9\u91cc\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>t<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e3a<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>m<\/span><\/mi>\u00d7<\/span><\/mo>t<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow>t<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e3a<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>t<\/span><\/mi>\u00d7<\/span><\/mo>t<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5bf9\u89d2\u77e9\u9635\uff0c<\/span><\/sub><\/span><\/mo><\/msubsup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e3a<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>t<\/span><\/mi>\u00d7<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u6211\u4eec\u53ea\u8ba1\u7b97<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684t\u4e2a\u5217\u5411\u91cf\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684t\u4e2a\u884c\u5411\u91cf\uff0c\u5b83\u4eec\u5bf9\u5e94<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03a3<\/span><\/mi><\/mrow>t<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4e2d\u6700\u5927\u7684t\u4e2a\u5947\u5f02\u503c\u3002\u5f53<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>t<\/span><\/mi>\u226a<\/span><\/mo>r<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u65f6\uff0c\u8fd9\u6bd4\u7d27\u51d1SVD\u66f4\u5feb\u66f4\u7701\uff0c\u4f46\u9700\u8981\u4e00\u5957\u5b8c\u5168\u4e0d\u540c\u7684\u6570\u503c\u6c42\u89e3\u5668\u5de5\u5177\u3002<\/span><\/section>\n

          <\/span><\/section>\n
          \u5728\u9700\u8981\u8fd1\u4f3c\u77e9\u9635<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684Moore\u2013Penrose\u9006\u7684\u5e94\u7528\u4e2d\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u6700\u5c0f\u5947\u5f02\u503c\u662f\u6709\u610f\u4e49\u7684\uff0c\u4f46\u4e0e\u6700\u5927\u5947\u5f02\u503c\u76f8\u6bd4\uff0c\u5b83\u4eec\u66f4\u96be\u8ba1\u7b97\u3002<\/span><\/section>\n

          <\/span><\/section>\n
          \u622a\u65adSVD\u5728\u6f5c\u5728\u8bed\u4e49\u7d22\u5f15\u4e2d\u5f97\u5230\u4e86\u5e94\u7528[30] \u3002<\/span><\/section>\n

          <\/span><\/section>\n

          <\/span><\/section>\n

          \n
          \n
          \n
          \n
          \n

          <\/section>\n

          <\/section>\n
          \n
          <\/svg><\/section>\n<\/section>\n<\/section>\n
          \n

          8. \u8303\u6570<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

          \n

          <\/section>\n

          <\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


          <\/p>\n

          8.1 Ky Fan\u8303\u6570<\/span><\/strong><\/span><\/h3>\n

          <\/span><\/section>\n
          \u6211\u4eec\u628a<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684k\u4e2a\u6700\u5927\u5947\u5f02\u503c\u4e4b\u548c\u79f0\u4e3a<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684Ky Fan k-\u8303\u6570\uff0c\u8fd9\u662f\u4e00\u79cd\u77e9\u9635\u8303\u6570\uff08matrix norm\uff09[31]\u3002<\/span><\/section>\n

          <\/span><\/section>\n
          Ky Fan\u8303\u6570\u4e2d\u7684\u7b2c\u4e00\u4e2a\uff0c\u5373Ky Fan 1-\u8303\u6570\uff0c\u7b49\u540c\u4e8e<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u4f5c\u4e3a\u7ebf\u6027\u7b97\u5b50\u76f8\u5bf9\u4e8e<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>K<\/span><\/mi>m<\/span><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u548c<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>K<\/span><\/mi>n<\/span><\/mi><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u6b27\u51e0\u91cc\u5f97\u8303\u6570\u7684\u7b97\u5b50\u8303\u6570\uff08operator norm\uff09\u3002\u6362\u8a00\u4e4b\uff0cKy Fan 1-\u8303\u6570\u5c31\u662f\u6807\u51c6<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u2113<\/span><\/mi>2<\/span><\/mn><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u6b27\u51e0\u91cc\u5f97\u5185\u79ef\u8bf1\u5bfc\u7684\u7b97\u5b50\u8303\u6570\u3002\u6211\u4eec\u5f88\u5bb9\u6613\u5c31\u80fd\u9a8c\u8bc1Ky Fan 1-\u8303\u6570\u548c\u5947\u5f02\u503c\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u4e00\u822c\u6765\u8bf4\uff0c\u5bf9\u4e8e\uff08\u53ef\u80fd\u662f\u65e0\u9650\u7ef4\uff09\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u4e0a\u7684\u6709\u754c\u7b97\u5b50<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u6211\u4eec\u6709\uff1a<\/span><\/section>\n

          <\/span><\/section>\n
          <\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-nstrut><\/mjx-c><\/mjx-mn><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mn><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-texatom><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/p>\n

          <\/span><\/mn><\/mfrac><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/span><\/section>\n
          \u5728\u77e9\u9635\u60c5\u51b5\u4e0b\uff0c<\/span><\/sub><\/span><\/mn><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u662f\u4e2a\u6b63\u89c4\u77e9\u9635\uff0c\u6240\u4ee5<\/span> <\/span> <\/sub><\/span><\/mn><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u5c31\u662f <\/span> <\/sub><\/span><\/mjx-container><\/span><\/mn><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u6700\u5927\u7279\u5f81\u503c\uff0c\u4e5f\u5c31\u662f<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u6700\u5927\u5947\u5f02\u503c\u3002<\/span><\/section>\n

          <\/span><\/section>\n
          Ky Fan\u8303\u6570\u4e2d\u7684\u6700\u540e\u4e00\u4e2a\uff0c\u5373\u6240\u6709\u5947\u5f02\u503c\u7684\u548c\uff0c\u6211\u4eec\u79f0\u4e4b\u4e3a\u8ff9\u8303\u6570<\/span>\uff08trace norm\uff09\uff08\u4e5f\u53eb\u6838\u8303\u6570\uff09<\/span>\uff0c\u5b9a\u4e49\u4e3a<\/span> <\/span><\/sub><\/mn><\/mrow><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff08<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u7279\u5f81\u503c\u662f\u5947\u5f02\u503c\u7684\u5e73\u65b9\uff09<\/span>\u3002<\/span><\/section>\n

          <\/span><\/section>\n

          8.2 \u5e0c\u5c14\u4f2f\u7279-\u65bd\u5bc6\u7279\u8303\u6570<\/span><\/strong><\/span><\/h3>\n

          <\/span><\/section>\n
          \u5947\u5f02\u503c\u8fd8\u4e0e\u7b97\u5b50\u7a7a\u95f4\u4e0a\u7684\u53e6\u4e00\u4e2a\u8303\u6570\u6709\u5173\u3002\u6211\u4eec\u6765\u770b\u770b<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>n<\/span><\/mi>\u00d7<\/span><\/mo>n<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u77e9\u9635\u4e0a\u7684\u5e0c\u5c14\u4f2f\u7279-\u65bd\u5bc6\u7279\u5185\u79ef<\/span>\uff08Hilbert\u2013Schmidt inner product\uff09<\/span>\uff0c\u5b83\u7684\u5b9a\u4e49\u662f\uff1a<\/span><\/section>\n

          <\/span><\/section>\n
          <\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/p>\n

          <\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n
          \u7531\u6b64\uff0c\u8bf1\u5bfc\u8303\u6570\u5c31\u662f\uff1a<\/span><\/section>\n

          <\/span><\/section>\n
          <\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-surd><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-surd><\/mjx-c><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/p>\n

          <\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/span><\/section>\n
          \u56e0\u4e3a\u8ff9\u5728\u9149\u7b49\u4ef7\u4e0b\u4e0d\u53d8\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ee5\u5f97\u51fa\uff1a<\/span><\/section>\n

          <\/span><\/section>\n
          <\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-surd><\/mjx-c><\/mjx-mo><\/mjx-base><\/mjx-row><\/mjx-c><\/mjx-mi><\/mjx-under><\/mjx-row><\/mjx-munder><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-spacer><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msubsup><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/p>\n

          <\/span><\/mn><\/msubsup><\/msqrt><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/span><\/mn><\/msubsup><\/msqrt><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n
          \u8fd9\u91cc<\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>i<\/span><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u662f<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684\u5947\u5f02\u503c\u3002\u6211\u4eec\u628a\u8fd9\u4e2a\u79f0\u4e3a<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684Frobenius\u8303\u6570\u3001Schatten 2-\u8303\u6570\u6216\u5e0c\u5c14\u4f2f\u7279-\u65bd\u5bc6\u7279\u8303\u6570\u3002\u76f4\u63a5\u8ba1\u7b97\u53ef\u4ee5\u53d1\u73b0\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>=<\/span><\/mo>(<\/span><\/mo>m<\/span><\/mi>i<\/span><\/sub><\/mi>j<\/span><\/sub><\/mi><\/mrow><\/msub>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u7684Frobenius\u8303\u6570\u4e0e\u4e0b\u5f0f\u4e00\u81f4\uff1a<\/span><\/section>\n
          <\/mjx-c><\/mjx-mo><\/mjx-surd><\/mjx-c><\/mjx-mo><\/mjx-base><\/mjx-row><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-under><\/mjx-row><\/mjx-munder><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msup><\/mjx-box><\/mjx-sqrt><\/mjx-msqrt><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/span><\/mo><\/munder><\/msqrt><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/p>\n

          <\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/span><\/section>\n
          \u503c\u5f97\u4e00\u63d0\u7684\u662f\uff0cFrobenius\u8303\u6570\u548c\u8ff9\u8303\u6570<\/span>\uff08\u6838\u8303\u6570\uff09<\/span>\u90fd\u662fSchatten\u8303\u6570\u7684\u7279\u6b8a\u60c5\u51b5\u3002<\/span><\/section>\n

          <\/span><\/section>\n

          <\/span><\/section>\n

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          \n
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          \n
          \n

          <\/section>\n

          <\/section>\n
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          <\/svg><\/section>\n<\/section>\n<\/section>\n
          \n

          9. \u53d8\u4f53\u548c\u63a8\u5e7f<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

          \n

          <\/section>\n

          <\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


          <\/p>\n

          9.1 \u5c3a\u5ea6\u4e0d\u53d8\u5947\u5f02\u503c\u5206\u89e3<\/span><\/strong><\/span><\/h3>\n

          <\/span><\/section>\n
          \u77e9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5947\u5f02\u503c\u662f\u552f\u4e00\u7684\uff0c\u4e14\u5bf9 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5de6\u53f3\u9149\u53d8\u6362\u4fdd\u6301\u4e0d\u53d8\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5bf9\u9149\u77e9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>A<\/span><\/mi><\/mrow>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5947\u5f02\u503c\u4e0e <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u76f8\u540c\u3002\u8fd9\u4e00\u91cd\u8981\u6027\u8d28\u9002\u7528\u4e8e\u9700\u8981\u4fdd\u6301\u6b27\u51e0\u91cc\u5f97\u8ddd\u79bb\u548c\u65cb\u8f6c\u4e0d\u53d8\u6027\u7684\u5e94\u7528\u3002<\/span><\/section>\n

          <\/span><\/section>\n
          \u5c3a\u5ea6\u4e0d\u53d8\u5947\u5f02\u503c\u5206\u89e3<\/span>\uff08SI-SVD\uff09<\/span>\u4e0e\u5e38\u89c4\u5947\u5f02\u503c\u5206\u89e3\u76f8\u4f3c\uff0c\u4f46\u5176\u5947\u5f02\u503c\u5bf9 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5bf9\u89d2\u53d8\u6362\u4fdd\u6301\u4e0d\u53d8\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5bf9\u53ef\u9006\u5bf9\u89d2\u77e9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>D<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>E<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>D<\/span><\/mi><\/mrow>A<\/span><\/mi><\/mrow>E<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5947\u5f02\u503c\u4e0e <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u76f8\u540c\u3002\u8fd9\u4e00\u6027\u8d28\u5728\u9700\u8981\u5bf9\u53d8\u91cf\u5355\u4f4d\u9009\u62e9<\/span>\uff08\u5982\u516c\u5236\u4e0e\u82f1\u5236\uff09<\/span>\u4fdd\u6301\u4e0d\u53d8\u7684\u5e94\u7528\u4e2d\u5c24\u4e3a\u91cd\u8981\u3002<\/span><\/section>\n


          <\/span><\/h3>\n

          9.2 \u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u4e0a\u7684\u6709\u754c\u7b97\u5b50<\/span><\/strong><\/span><\/h3>\n

          <\/span><\/section>\n
          \u6211\u4eec\u53ef\u4ee5\u5c06\u5206\u89e3 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>=<\/span><\/mo>U<\/span><\/mi><\/mrow>\u03a3<\/span><\/mi><\/mrow>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u63a8\u5e7f\u5230\u53ef\u5206\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>H<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e0a\u7684\u6709\u754c\u7b97\u5b50<\/span>\uff08bounded operator\uff09<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5bf9\u4efb\u4f55\u6709\u754c\u7b97\u5b50 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u90fd\u5b58\u5728\u90e8\u5206\u7b49\u8ddd<\/span>\uff08partial isometry\uff09<\/span>\u7b97\u5b50 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u9149\u7b97\u5b50 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u6d4b\u5ea6\u7a7a\u95f4 <\/span><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>(<\/span><\/mo>X<\/span><\/mi>,<\/span><\/mo>\u03bc<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u4ee5\u53ca\u975e\u8d1f\u53ef\u6d4b\u51fd\u6570 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>f<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u4f7f\u5f97\uff1a<\/span><\/section>\n

          <\/span><\/section>\n
          <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/p>\n

          <\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/span><\/section>\n
          \u8fd9\u91cc <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>T<\/span><\/mi>f<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>L<\/span><\/mi>2<\/span><\/sup><\/mn><\/msup>(<\/span><\/mo>X<\/span><\/mi>,<\/span><\/mo>\u03bc<\/span><\/mi>)<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e0a\u7684 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>f<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e58\u6cd5\u7b97\u5b50\u3002<\/span><\/section>\n

          <\/span><\/section>\n
          \u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u6a21\u4eff\u4e0a\u8ff0\u77e9\u9635\u60c5\u51b5\u7684\u7ebf\u6027\u4ee3\u6570\u8bba\u8bc1\u6765\u8bc1\u660e\u8fd9\u4e00\u70b9\u3002<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>T<\/span><\/mi>f<\/span><\/sub><\/mi><\/msub>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> <\/span><\/sup>\u662f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u552f\u4e00\u6b63\u5e73\u65b9\u6839\uff0c\u8fd9\u7531\u81ea\u4f34\u7b97\u5b50<\/span>\uff08self-adjoint operators\uff09<\/span>\u7684Borel\u51fd\u6570\u6f14\u7b97<\/span> (Borel functional calculus) <\/span>\u7ed9\u51fa\u3002<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e0d\u5fc5\u662f\u9149\u7684\u539f\u56e0\u662f\uff0c\u4e0e\u6709\u9650\u7ef4\u60c5\u51b5\u4e0d\u540c\uff0c\u7ed9\u5b9a\u5177\u6709\u975e\u5e73\u51e1\u6838\u7684\u7b49\u8ddd\u7b97\u5b50 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi>1<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u53ef\u80fd\u627e\u4e0d\u5230\u5408\u9002\u7684 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi>2<\/span><\/sub><\/mn><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4f7f\u5f97\uff1a<\/span><\/section>\n

          <\/span><\/section>\n
          <\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msub><\/mjx-tstrut><\/mjx-mtd><\/mjx-mtr><\/mjx-itable><\/mjx-table><\/mjx-mtable><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/span><\/mo><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n
          <\/section>\n
          <\/span><\/mo><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/span><\/section>\n
          \u6210\u4e3a\u4e00\u4e2a\u9149\u7b97\u5b50\u3002<\/span><\/section>\n

          <\/span><\/section>\n
          \u4e0e\u77e9\u9635\u7c7b\u4f3c\uff0c\u5947\u5f02\u503c\u5206\u89e3\u7b49\u4ef7\u4e8e\u7b97\u5b50\u7684\u6781\u5206\u89e3\uff1a\u6211\u4eec\u53ef\u4ee5\u7b80\u5355\u5730\u5199\u4f5c\uff1a<\/span><\/section>\n

          <\/span><\/section>\n
          <\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math><\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/p>\n

          <\/span><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/span><\/section>\n
          \u5e76\u6ce8\u610f\u5230 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>U<\/span><\/mi><\/mrow>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4ecd\u662f\u90e8\u5206\u7b49\u8ddd\u7b97\u5b50\uff0c\u800c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>V<\/span><\/mi><\/mrow>T<\/span><\/mi>f<\/span><\/sub><\/mi><\/msub>V<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u6b63\u7684\u3002<\/span><\/section>\n


          <\/span><\/h3>\n

          9.3 \u5947\u5f02\u503c\u548c\u7d27\u7b97\u5b50<\/span><\/strong><\/span><\/h3>\n

          <\/span><\/section>\n
          \u6211\u4eec\u53ef\u4ee5\u5c06\u5947\u5f02\u503c\u548c\u5de6\/\u53f3\u5947\u5f02\u5411\u91cf\u7684\u6982\u5ff5\u63a8\u5e7f\u5230\u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u4e0a\u7684\u7d27\u7b97\u5b50<\/span>\uff08compact operator on Hilbert space\uff09<\/span>\uff0c\u56e0\u4e3a\u5b83\u4eec\u5177\u6709\u79bb\u6563\u8c31\u3002\u5982\u679c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>T<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u7d27\u7684\uff0c\u5176\u8c31\u4e2d\u7684\u6bcf\u4e2a\u975e\u96f6 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03bb<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5c31\u90fd\u662f\u7279\u5f81\u503c\u3002\u6b64\u5916\uff0c\u7d27\u81ea\u4f34\u7b97\u5b50\u53ef\u7531\u5176\u7279\u5f81\u5411\u91cf\u5bf9\u89d2\u5316\u3002\u5982\u679c <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u7d27\u7684\uff0c\u90a3\u4e48 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/mo><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e5f\u662f\u7d27\u7684\u3002\u5e94\u7528\u5bf9\u89d2\u5316\u7ed3\u679c\uff0c\u5176\u6b63\u5e73\u65b9\u6839 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>T<\/span><\/mi>f<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u9149\u50cf\u6709\u4e00\u7ec4\u5bf9\u5e94\u4e8e\u4e25\u683c\u6b63\u7279\u5f81\u503c <\/span><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>{<\/span><\/mo>\u03c3<\/span><\/mi>i<\/span><\/sub><\/mi><\/msub>}<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u6b63\u4ea4\u5f52\u4e00\u5316\u7279\u5f81\u5411\u91cf <\/span><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>{<\/span><\/mo>e<\/span><\/mi>i<\/span><\/sub><\/mi><\/msub>}<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\u3002\u5bf9 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>H<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e2d\u7684\u4efb\u610f <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c8<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container>\uff0c\u6709\uff1a<\/span><\/section>\n

          <\/span><\/section>\n
          <\/span><\/section>\n

          <\/p>\n

          <\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container><\/section>\n

          <\/span><\/section>\n
          \u8fd9\u91cc\u7ea7\u6570\u5728 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>H<\/span><\/mi><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u8303\u6570\u62d3\u6251\u4e2d\u6536\u655b\u3002\u6ce8\u610f\uff0c\u8fd9\u4e0e\u6709\u9650\u7ef4\u60c5\u51b5\u7684\u8868\u8fbe\u5f0f\u76f8\u4f3c\u3002\u6211\u4eec\u79f0 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>i<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5947\u5f02\u503c\uff0c<\/span><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>{<\/span><\/mo>U<\/span><\/mi><\/mrow>e<\/span><\/mi>i<\/span><\/sub><\/mi><\/msub>}<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u548c <\/span><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>{<\/span><\/mo>U<\/span><\/mi><\/mrow>e<\/span><\/mi>i<\/span><\/sub><\/mi><\/msub>}<\/span><\/mo><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u5206\u522b\u4e3a <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u5de6\u5947\u5f02\u548c\u53f3\u5947\u5f02\u5411\u91cf\u3002<\/span><\/section>\n

          <\/span><\/section>\n
          \u5e0c\u5c14\u4f2f\u7279\u7a7a\u95f4\u4e0a\u7684\u7d27\u7b97\u5b50\u662f\u6709\u9650\u79e9\u7b97\u5b50<\/span>\uff08finite rank operator\uff09<\/span>\u5728\u4e00\u81f4\u7b97\u5b50\u62d3\u6251\u4e2d\u7684\u95ed\u5305\u3002\u4e0a\u8ff0\u7ea7\u6570\u8868\u8fbe\u5f0f\u7ed9\u51fa\u4e86\u8fd9\u79cd\u8868\u793a\u7684\u4e00\u4e2a\u660e\u786e\u4f8b\u5b50\u3002\u8fd9\u76f4\u63a5\u5bfc\u51fa\u4ee5\u4e0b\u7ed3\u8bba\uff1a<\/span><\/section>\n

          <\/span><\/section>\n
          \u5b9a\u7406\uff1a<\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u7d27\u7684\u5f53\u4e14\u4ec5\u5f53 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-script><\/mjx-msup><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>M<\/span><\/mi><\/mrow>\u2217<\/span><\/sup><\/mo><\/msup>M<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u662f\u7d27\u7684\u3002<\/span><\/section>\n

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          <\/svg><\/section>\n<\/section>\n<\/section>\n
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          10. \u5386\u53f2<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

          \n

          <\/section>\n

          <\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


          <\/p>\n

          \u5947\u5f02\u503c\u5206\u89e3\u6700\u521d\u6e90\u4e8e\u5fae\u5206\u51e0\u4f55\u5b66\u5bb6\u7684\u7814\u7a76\u3002\u4ed6\u4eec\u5e0c\u671b\u786e\u5b9a\u80fd\u5426\u901a\u8fc7\u5bf9\u4e24\u4e2a\u4f5c\u7528\u7a7a\u95f4\u8fdb\u884c\u72ec\u7acb\u6b63\u4ea4\u53d8\u6362\uff0c\u4f7f\u4e00\u4e2a\u5b9e\u53cc\u7ebf\u6027\u5f62\u5f0f\u7b49\u540c\u4e8e\u53e6\u4e00\u4e2a\u30021873\u5e74\u548c1874\u5e74\uff0cEugenio Beltrami \u548c Camille Jordan \u5206\u522b\u72ec\u7acb\u53d1\u73b0\uff0c\u53cc\u7ebf\u6027\u5f62\u5f0f<\/span>\uff08\u7528\u77e9\u9635\u8868\u793a\uff09<\/span>\u7684\u5947\u5f02\u503c\u6784\u6210\u4e86\u6b63\u4ea4\u66ff\u4ee3\u4e0b\u7684\u5b8c\u6574\u4e0d\u53d8\u91cf\u96c6<\/span>\uff08complete set of invariants\uff09<\/span>\u30021889\u5e74\uff0cJames Joseph Sylvester \u4e5f\u72ec\u7acb\u5f97\u51fa\u4e86\u5b9e\u65b9\u9635\u7684\u5947\u5f02\u503c\u5206\u89e3\uff0c\u4ed6\u5c06\u5947\u5f02\u503c\u79f0\u4e3a\u77e9\u9635 <\/span><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>A<\/span><\/mi><\/mrow><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u7684\u89c4\u8303\u4e58\u5b50\u30021915\u5e74\uff0cAutonne \u6210\u4e3a\u7b2c\u56db\u4f4d\u72ec\u7acb\u53d1\u73b0\u8005\uff0c\u4ed6\u901a\u8fc7\u6781\u5206\u89e3\u63a8\u5bfc\u51fa\u4e86\u5947\u5f02\u503c\u5206\u89e3\u3002\u76f4\u52301936\u5e74\uff0cCarl Eckart \u548c Gale J. Young \u624d\u9996\u6b21\u8bc1\u660e\u4e86\u77e9\u5f62\u548c\u590d\u77e9\u9635\u7684\u5947\u5f02\u503c\u5206\u89e3[32]\uff0c\u4ed6\u4eec\u5c06\u5176\u89c6\u4e3a\u5384\u7c73\u77e9\u9635\u4e3b\u8f74\u53d8\u6362\u7684\u63a8\u5e7f\u3002<\/span><\/section>\n

          <\/span><\/section>\n
          1907\u5e74\uff0cErhard Schmidt \u4e3a\u79ef\u5206\u7b97\u5b50<\/span>\uff08integral operators\uff09\uff08\u5728\u67d0\u4e9b\u5f31\u6280\u672f\u5047\u8bbe\u4e0b\u4e3a\u7d27\u7b97\u5b50\uff09<\/span>\u5b9a\u4e49\u4e86\u7c7b\u4f3c\u5947\u5f02\u503c\u7684\u6982\u5ff5\uff0c\u4ed6\u4f3c\u4e4e\u4e0d\u77e5\u9053\u6709\u5173\u6709\u9650\u77e9\u9635\u5947\u5f02\u503c\u7684\u5e73\u884c\u7814\u7a76\u30021910\u5e74\uff0c\u00c9mile Picard \u8fdb\u4e00\u6b65\u53d1\u5c55\u4e86\u8fd9\u4e00\u7406\u8bba\uff0c\u5e76\u9996\u6b21\u5c06 <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-mstyle><\/mjx-math>\u03c3<\/span><\/mi>k<\/span><\/sub><\/mi><\/msub><\/mrow><\/mstyle><\/mjx-assistive-mml><\/mjx-container> \u79f0\u4e3a\u5947\u5f02\u503c\u3002<\/span><\/section>\n

          <\/span><\/section>\n
          \u8ba1\u7b97\u5947\u5f02\u503c\u5206\u89e3\u7684\u5b9e\u7528\u65b9\u6cd5\u53ef\u8ffd\u6eaf\u52301954-1955\u5e74 Kogbetliantz \u548c1958\u5e74 Hestenes \u7684\u5de5\u4f5c[33]\u3002\u8fd9\u4e9b\u65b9\u6cd5\u4e0e\u4f7f\u7528\u5e73\u9762\u65cb\u8f6c\u6216 Givens\u65cb\u8f6c\u7684 Jacobi \u7279\u5f81\u503c\u7b97\u6cd5\u6781\u4e3a\u76f8\u4f3c\u3002\u7136\u800c\uff0c1965\u5e74 Gene Golub \u548c William Kahan \u53d1\u8868\u7684\u65b9\u6cd5[23]\u53d6\u4ee3\u4e86\u4e4b\u524d\u7684\u65b9\u6cd5\uff0c\u4ed6\u4eec\u91c7\u7528\u4e86 Householder \u53d8\u6362\u6216\u53cd\u5c04\u30021970\u5e74\uff0cGolub \u548c Christian Reinsch[34] \u53d1\u8868\u4e86 Golub\/Kahan \u7b97\u6cd5\u7684\u4e00\u4e2a\u53d8\u4f53\uff0c\u8fd9\u4e2a\u53d8\u4f53\u81f3\u4eca\u4ecd\u5e7f\u6cdb\u5e94\u7528\u3002<\/span><\/section>\n

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          \u53c2\u8003\u6587\u732e<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n

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            \n
          1. \n

            Stewart, G. W. (1993). “On the Early History of the Singular Value Decomposition”. SIAM Review. 35 (4): 551\u2013566. CiteSeerX 10.1.1.23.1831. doi:10.1137\/1035134. hdl:1903\/566. JSTOR 2132388.<\/p>\n<\/li>\n

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             DeAngelis, G. C.; Ohzawa, I.; Freeman, R. D. (October 1995), “”Receptive-field dynamics in the central visual pathways””, Trends Neurosci, 18 (10): 451\u20138., doi:10.1016\/0166-2236(95)94496-R, PMID 8545912, S2CID 12827601<\/p>\n<\/li>\n

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             Depireux, D. A.; Simon, J. Z.; Klein, D. J.; Shamma, S. A. (March 2001), “”Spectro-temporal response field characterization with dynamic ripples in ferret primary auditory cortex””, J. Neurophysiol, 85 (3): 1220\u201334, doi:10.1152\/jn.2001.85.3.1220, PMID 11247991<\/p>\n<\/li>\n

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             The Singular Value Decomposition in Symmetric (Lowdin) Orthogonalization and Data Compression (PDF)<\/p>\n<\/li>\n

          5. \n

             Sahidullah, Md.; Kinnunen; Tomi (March 2016), “”Local spectral variability features for speaker verification””, Digital Signal Processing, 50: 1\u201311, doi:10.1016\/j.dsp.2015.10.011<\/p>\n<\/li>\n

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             Mademlis, Ioannis; Tefas, Anastasios; Pitas, Ioannis (2018), “Regularized SVD-Based Video Frame Saliency for Unsupervised Activity Video Summarization”, IEEE, pp. 2691\u20132695, doi:10.1109\/ICASSP.2018.8462274, ISBN 978-1-5386-4658-8, S2CID 52286352, retrieved 2023-01-19<\/p>\n<\/li>\n

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             Alter, O.; Brown, P. O.; Botstein, D. (September 2000). “Singular Value Decomposition for Genome-Wide Expression Data Processing and Modeling”. PNAS. 97 (18): 10101\u201310106. Bibcode:2000PNAS…9710101A. doi:10.1073\/pnas.97.18.10101. PMC 27718. PMID 10963673.<\/p>\n<\/li>\n

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             Alter, O.; Golub, G. H. (November 2004). “Integrative Analysis of Genome-Scale Data by Using Pseudoinverse Projection Predicts Novel Correlation Between DNA Replication and RNA Transcription”. PNAS. 101 (47): 16577\u201316582. Bibcode:2004PNAS..10116577A. doi:10.1073\/pnas.0406767101. PMC 534520. PMID 15545604.<\/p>\n<\/li>\n

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             Alter, O.; Golub, G. H. (August 2006). “Singular Value Decomposition of Genome-Scale mRNA Lengths Distribution Reveals Asymmetry in RNA Gel Electrophoresis Band Broadening”. PNAS. 103 (32): 11828\u201311833. Bibcode:2006PNAS..10311828A. doi:10.1073\/pnas.0604756103. PMC 1524674. PMID 16877539.<\/p>\n<\/li>\n

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             Bertagnolli, N. M.; Drake, J. A.; Tennessen, J. M.; Alter, O. (November 2013). “SVD Identifies Transcript Length Distribution Functions from DNA Microarray Data and Reveals Evolutionary Forces Globally Affecting GBM Metabolism”. PLOS ONE. 8 (11): e78913. Bibcode:2013PLoSO…878913B. doi:10.1371\/journal.pone.0078913. PMC 3839928. PMID 24282503.Highlight}<\/p>\n<\/li>\n

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             Edelman, Alan (1992), “”On the distribution of a scaled condition number”” (PDF), Math. Comp., 58 (197): 185\u2013190, Bibcode:1992MaCom..58..185E, doi:10.1090\/S0025-5718-1992-1106966-2<\/p>\n<\/li>\n

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             Shen, Jianhong (Jackie) (2001), “”On the singular values of Gaussian random matrices””, Linear Alg. Appl., 326 (1\u20133): 1\u201314, doi:10.1016\/S0024-3795(00)00322-0<\/p>\n<\/li>\n

          13. \n

             Walton, S.; Hassan, O.; Morgan, K. (2013), “”Reduced order modelling for unsteady fluid flow using proper orthogonal decomposition and radial basis functions””, Applied Mathematical Modelling, 37 (20\u201321): 8930\u20138945, doi:10.1016\/j.apm.2013.04.025<\/p>\n<\/li>\n

          14. \n

             Setyawati, Y.; Ohme, F.; Khan, S. (2019), “”Enhancing gravitational waveform model through dynamic calibration””, Physical Review D, 99 (2): 024010, arXiv:1810.07060, Bibcode:2019PhRvD..99b4010S, doi:10.1103\/PhysRevD.99.024010, S2CID 118935941<\/p>\n<\/li>\n

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             Sarwar, Badrul; Karypis, George; Konstan, Joseph A.; Riedl, John T. (2000), “Application of Dimensionality Reduction in Recommender System \u2013 A Case Study” (PDF), University of Minnesota<\/p>\n<\/li>\n

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             Bosagh Zadeh, Reza; Carlsson, Gunnar (2013), “Dimension Independent Matrix Square Using MapReduce” (PDF), arXiv:1304.1467, Bibcode:2013arXiv1304.1467B<\/p>\n<\/li>\n

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             Fanaee Tork, Hadi; Gama, Jo\u00e3o (September 2014), “”Eigenspace method for spatiotemporal hotspot detection””, Expert Systems, 32 (3): 454\u2013464, arXiv:1406.3506, Bibcode:2014arXiv1406.3506F, doi:10.1111\/exsy.12088, S2CID 15476557<\/p>\n<\/li>\n

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             Fanaee Tork, Hadi; Gama, Jo\u00e3o (May 2015), “”EigenEvent: An Algorithm for Event Detection from Complex Data Streams in Syndromic Surveillance””, Intelligent Data Analysis, 19 (3): 597\u2013616, arXiv:1406.3496, doi:10.3233\/IDA-150734, S2CID 17966555<\/p>\n<\/li>\n

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             Muralidharan, Vivek; Howell, Kathleen (2023), “”Stretching directions in cislunar space: Applications for departures and transfer design””, Astrodynamics, 7 (2): 153\u2013178, Bibcode:2023AsDyn…7..153M, doi:10.1007\/s42064-022-0147-z<\/p>\n<\/li>\n

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             Muralidharan, Vivek; Howell, Kathleen (2022), “”Leveraging stretching directions for stationkeeping in Earth-Moon halo orbits””, Advances in Space Research, 69 (1): 620\u2013646, Bibcode:2022AdSpR..69..620M, doi:10.1016\/j.asr.2021.10.028, S2CID 239490016<\/p>\n<\/li>\n

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             Rijk, P.P.M. de (1989). “A one-sided Jacobi algorithm for computing the singular value decomposition on a vector computer”. SIAM J. Sci. Stat. Comput. 10: 359.<\/p>\n<\/li>\n

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             Golub, Gene H.; Kahan, William (1965). “Calculating the singular values and pseudo-inverse of a matrix”. Journal of the Society for Industrial and Applied Mathematics, Series B: Numerical Analysis. 2 (2): 205\u2013224. Bibcode:1965SJNA….2..205G. doi:10.1137\/0702016. JSTOR 2949777.<\/p>\n<\/li>\n

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             Demmel, James; Kahan, William (1990). “Accurate singular values of bidiagonal matrices”. SIAM Journal on Scientific and Statistical Computing. 11 (5): 873\u2013912. CiteSeerX 10.1.1.48.3740. doi:10.1137\/0911052.<\/p>\n<\/li>\n

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             Demmel, James (2000). “Decompositions”. Templates for the Solution of Algebraic Eigenvalue Problems. By Bai, Zhaojun; Demmel, James; Dongarra, Jack J.; Ruhe, Axel; van der Vorst, Henk A.. Society for Industrial and Applied Mathematics. doi:10.1137\/1.9780898719581. ISBN 978-0-89871-471-5.<\/p>\n<\/li>\n

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             Chicco, D; Masseroli, M (2015). “Software suite for gene and protein annotation prediction and similarity search”. IEEE\/ACM Transactions on Computational Biology and Bioinformatics. 12 (4): 837\u2013843. doi:10.1109\/TCBB.2014.2382127. hdl:11311\/959408. PMID 26357324. S2CID 14714823.<\/p>\n<\/li>\n

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             Eckart, C.; Young, G. (1936). “The approximation of one matrix by another of lower rank”. Psychometrika. 1 (3): 211\u20138. doi:10.1007\/BF02288367. S2CID 10163399.<\/p>\n<\/li>\n

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             Hestenes, M. R. (1958). “Inversion of Matrices by Biorthogonalization and Related Results”. Journal of the Society for Industrial and Applied Mathematics. 6 (1): 51\u201390. doi:10.1137\/0106005. JSTOR 2098862. MR 0092215.<\/p>\n<\/li>\n

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             Golub, G. H.; Reinsch, C. (1970). “Singular value decomposition and least squares solutions”. Numerische Mathematik. 14 (5): 403\u2013420. doi:10.1007\/BF02163027. MR 1553974. S2CID 123532178.<\/p>\n<\/li>\n<\/ol>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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            \uff08\u53c2\u8003\u6587\u732e\u53ef\u4e0a\u4e0b\u6ed1\u52a8<\/strong>\u67e5\u770b\uff09<\/p>\n<\/section>\n<\/section>\n


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            \u8de8\u5c3a\u5ea6\u3001\u8de8\u5c42\u6b21\u7684\u6d8c\u73b0\u662f\u590d\u6742\u7cfb\u7edf\u7814\u7a76\u7684\u5173\u952e\u95ee\u9898\uff0c\u751f\u547d\u8d77\u6e90\u548c\u610f\u8bc6\u8d77\u6e90\u8fd9\u4e24\u5ea7\u4ef0\u4e4b\u5f25\u9ad8\u7684\u5927\u5c71\u662f\u5176\u4ee3\u8868\u3002\u4ece2021\u5e74\u590f\u5929\u81f3\u4eca\uff0c\u96c6\u667a\u4ff1\u4e50\u90e8\u5df2\u7ecf\u9646\u7eed\u4e3e\u529e\u4e86\u56db\u5b63\u300c\u56e0\u679c\u6d8c\u73b0\u300d\u8bfb\u4e66\u4f1a\uff0c\u7cfb\u7edf\u68b3\u7406\u4e86\u56e0\u679c\u6d8c\u73b0\u7406\u8bba\u7684\u53d1\u5c55\u8109\u7edc\uff0c\u6df1\u5165\u63a2\u8ba8\u4e86\u4fe1\u606f\u6574\u5408\u4e0e\u4fe1\u606f\u5206\u89e3\u7684\u672c\u8d28\uff0c\u5e76\u63a2\u7d22\u4e86\u5728\u751f\u7269\u7f51\u7edc\u3001\u8111\u7f51\u7edc\u3001\u673a\u5668\u5b66\u4e60\u7b49\u8de8\u5b66\u79d1\u9886\u57df\u7684\u5e94\u7528\u3002\u6b64\u6b21\u56e0\u679c\u6d8c\u73b0\u8bfb\u4e66\u4f1a\u7b2c\u4e94\u5b63<\/a>\u5c06\u8ffd\u8e2a\u56e0\u679c\u6d8c\u73b0\u9886\u57df\u7684\u524d\u6cbf\u8fdb\u5c55\uff0c\u5c55\u793a\u96c6\u667a\u793e\u533a\u6210\u5458\u7684\u539f\u521b\u6027\u5de5\u4f5c\uff0c\u5e0c\u671b\u63a2\u8ba8\u56e0\u679c\u6d8c\u73b0\u7406\u8bba\u3001\u590d\u6742\u7cfb\u7edf\u7684\u4f4e\u79e9\u8868\u793a\u7406\u8bba\u3001\u672c\u5f81\u5fae\u89c2\u6001\u7406\u8bba\u4e4b\u95f4\u7684\u76f8\u901a\u4e4b\u5904\uff0c\u5bf9\u590d\u6742\u7cfb\u7edf\u7684\u6d8c\u73b0\u73b0\u8c61\u6709\u66f4\u6df1\u523b\u7684\u7406\u89e3\u3002\u8bfb\u4e66\u4f1a\u5df2\u5b8c\u7ed3\uff0c\u73b0\u5728\u62a5\u540d\u53ef\u52a0\u5165\u793e\u7fa4\u5e76\u89e3\u9501\u56de\u653e\u89c6\u9891\u6743\u9650\u3002<\/span><\/span><\/p>\n


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            \u201c\u590d\u6742 AI \u6b21\u65b9\u201d\u5f00\u653e\u5b9e\u9a8c\u5ba4\u62db\u52df<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n


            <\/section>\n

            \u4f5c\u4e3a\u5317\u5e08\u5927\u7cfb\u7edf\u79d1\u5b66\u5b66\u9662\u6559\u6388\u3001\u96c6\u667a\u4ff1\u4e50\u90e8\u4e0e\u96c6\u667a\u5b66\u56ed\u521b\u59cb\u4eba\u3001\u96c6\u667a\u79d1\u5b66\u7814\u7a76\u4e2d\u5fc3\u9662\u957f\uff0c\u5f20\u6c5f\u4ece2003\u5e74\u5f00\u59cb\u957f\u671f\u4ece\u4e8b\u6709\u5173\u590d\u6742\u7cfb\u7edf\u5efa\u6a21\u7684\u5de5\u4f5c\u3002\u8fd1\u5e74\u6765\uff0c\u5f20\u6c5f\u5e26\u9886\u7740\u5317\u5e08\u5927\u7684\u7814\u7a76\u7ec4\u5f00\u59cb\u805a\u7126\u5728\u57fa\u4e8e\u65b0\u5174AI\u6280\u672f\u8fdb\u884c\u57fa\u4e8e\u6570\u636e\u9a71\u52a8\u7684\u81ea\u52a8\u5efa\u6a21\u7814\u7a76\uff0c\u5e76\u7acb\u5fd7\u7834\u89e3\u590d\u6742\u7cfb\u7edf\u7684\u6d8c\u73b0\u4e4b\u8c1c\u3002\u6211\u4eec\u5e0c\u671b\u53ef\u4ee5\u6709\u5bf9\u590d\u6742\u7cfb\u7edf\u81ea\u52a8\u5efa\u6a21\u9886\u57df\u6709\u70ed\u60c5\uff0c\u4e14\u8ba4\u53ef\u8fd9\u4e2a\u9886\u57df\u53d1\u5c55\u524d\u666f\u7684\u670b\u53cb\u4e00\u8d77\u6765\u5408\u4f5c\uff0c\u4fc3\u8fdb\u8fd9\u4e00\u9886\u57df\u7684\u5feb\u901f\u53d1\u5c55\u3002<\/span>\u6211\u4eec\u5e0c\u671b\u8fd9\u4e2a\u53eb\u505a\u201c Complexity <\/span>AI<\/sup> \u201d\uff0c\u4e2d\u6587\u53eb\u505a\u201c\u590d\u6742AI\u6b21\u65b9\u201d\u7684\u5f00\u653e\u5b9e\u9a8c\u5ba4\uff0c\u80fd\u591f\u771f\u6b63\u5b9e\u73b0\u601d\u60f3\u5171\u4eab\u3001\u8d44\u6e90\u5171\u4eab\u3001\u8de8\u5b66\u79d1\u4ea4\u53c9\uff0c\u5171\u540c\u4e3a\u590d\u6742\u7cfb\u7edf\u81ea\u52a8\u5efa\u6a21\u800c\u594b\u8fdb\u3002<\/span><\/p>\n


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            \u8be6\u60c5\u8bf7\u89c1\uff1a<\/span>\u201c\u590d\u6742 AI \u6b21\u65b9\u201d\u5f00\u653e\u5b9e\u9a8c\u5ba4\u62db\u52df\uff0c\u6311\u6218\u201c\u6d8c\u73b0\u201d\u96be\u9898<\/a><\/span><\/p>\n


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            1. <\/strong>\u56e0\u679c\u6d8c\u73b0\uff1a\u7528\u56e0\u679c\u91cf\u5316\u590d\u6742\u7cfb\u7edf\u4e2d\u7684\u6d8c\u73b0\uff5c\u96c6\u667a\u767e\u79d1<\/a><\/strong><\/span><\/section>\n
            2. <\/strong>\u673a\u5668\u5b66\u4e60\u6846\u67b6NIS+\uff1a\u901a\u8fc7\u6700\u5927\u5316\u6709\u6548\u4fe1\u606f\u8bc6\u522b\u201c\u56e0\u679c\u6d8c\u73b0\u201d\uff5c\u96c6\u667a\u767e\u79d1<\/a><\/strong><\/span><\/section>\n
            3. <\/strong>\u5168\u7f51\u6700\u5168\u603b\u7ed3\uff01\u56e0\u679c\u6d8c\u73b0\u6838\u5fc3\u6307\u6807\u201c\u6709\u6548\u4fe1\u606f\u201d\uff5c\u96c6\u667a\u767e\u79d1<\/a><\/strong><\/span><\/section>\n
            4. <\/strong><\/span>\u5f20\u6c5f\uff1a\u7b2c\u4e09\u4ee3\u4eba\u5de5\u667a\u80fd\u6280\u672f\u57fa\u7840\u2014\u2014\u4ece\u53ef\u5fae\u5206\u7f16\u7a0b\u5230\u56e0\u679c\u63a8\u7406 | \u96c6\u667a\u5b66\u56ed\u5168\u65b0\u8bfe\u7a0b<\/a><\/strong><\/strong><\/span><\/section>\n
            <\/strong>5.<\/strong> \u9f99\u5e74\u5927\u8fd0\u8d77\uff0c\u5b66\u4e60\u6b63\u5f53\u65f6\uff01\u89e3\u9501\u96c6\u667a\u5168\u7ad9\u5185\u5bb9\uff0c\u5f00\u542f\u65b0\u5e74\u5b66\u4e60\u8ba1\u5212<\/a><\/strong><\/strong><\/strong><\/strong><\/span><\/strong><\/strong><\/section>\n

            6. <\/strong>\u52a0\u5165\u96c6\u667a\uff0c\u4e00\u8d77\u590d\u6742\uff01<\/strong><\/span><\/a><\/p>\n


            <\/p>\n


            <\/p>\n

            \u70b9\u51fb\u201c\u9605\u8bfb\u539f\u6587\u201d\uff0c\u62a5\u540d\u8bfb\u4e66\u4f1a<\/strong><\/span><\/strong><\/span><\/strong><\/p>\n

            <\/mp-style-type><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

            \u5bfc\u8bed \u5728\u7ebf\u6027\u4ee3\u6570\u4e2d\uff0c\u5947\u5f02\u503c\u5206\u89e3\uff08Singular value decomposition\uff09\u662f\u4e00\u79cd\u901a\u8fc7\u65cb\u8f6c\u3001\u7f29\u653e\u548c\u518d\u6b21\u65cb\u8f6c\u6765\u5bf9\u5b9e\u77e9\u9635\uff08real matrix \u5143\u7d20\u90fd\u662f\u5b9e\u6570\u7684\u77e9\u9635\uff09\u6216\u590d\u77e9\u9635\uff08complex matrix \u5143\u7d20\u6709\u590d\u6570\u7684\u77e9\u9635\uff09\u8fdb\u884c\u56e0\u5f0f\u5206\u89e3\u7684\u65b9\u6cd5\uff0c\u5982\u4e0b\u56fe\u3002\u5b83\u628a\u5177\u6709\u6807\u51c6\u6b63\u4ea4\u57fa\uff08orthonorm…<\/p>\n","protected":false},"author":0,"featured_media":55001,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"special":[],"_links":{"self":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts\/55080"}],"collection":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"replies":[{"embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=55080"}],"version-history":[{"count":0,"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts\/55080\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/media\/55001"}],"wp:attachment":[{"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=55080"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=55080"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=55080"},{"taxonomy":"special","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fspecial&post=55080"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}