{"id":549,"date":"2021-05-16T20:26:47","date_gmt":"2021-05-16T12:26:47","guid":{"rendered":"http:\/\/www.wayln.com\/?p=549"},"modified":"2021-05-16T21:01:27","modified_gmt":"2021-05-16T13:01:27","slug":"5-3-%e5%af%b9%e8%a7%92%e5%8c%96","status":"publish","type":"post","link":"https:\/\/www.wayln.com\/?p=549","title":{"rendered":"5.3 \u5bf9\u89d2\u5316"},"content":{"rendered":"<div id=\"toc_container\" class=\"toc_transparent no_bullets\"><p class=\"toc_title\">Contents<\/p><ul class=\"toc_list\"><li><a href=\"#i\"><span class=\"toc_number toc_depth_1\">1<\/span> \u77e9\u9635\u7684\u5bf9\u89d2\u5316<\/a><\/li><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_1\">2<\/span> \u7279\u5f81\u503c\u4e0d\u662f\u90fd\u76f8\u5f02\u7684\u77e9\u9635<\/a><\/li><\/ul><\/div>\n<p>\u5728\u5f88\u591a\u60c5\u51b5\u4e0b\uff0c\u4ece\u5206\u89e3\u5f0f<span class=\"katex math inline\">A=PDP^{-1}<\/span>,\u6211\u4eec\u80fd\u591f\u4e86\u89e3\u5230\u6709\u5173\u77e9\u9635A\u7684\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf\u7684\u4fe1\u606f\u3002<\/p>\n<p>\u5982\u679c\u65b9\u9635\u76f8\u4f3c\u4e8e\u5bf9\u89d2\u77e9\u9635\uff0c\u5373\u5b58\u5728\u53ef\u9006\u77e9\u9635P\u548c\u5bf9\u89d2\u77e9\u9635D\uff0c\u6709<span class=\"katex math inline\">A=PDP^{-1}<\/span>,\u5219\u79f0A<strong>\u53ef\u5bf9\u89d2\u5316<\/strong>\u3002<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u74065<\/strong> \uff08\u5bf9\u89d2\u5316\u5b9a\u7406\uff09<br \/>\n  nxn\u77e9\u9635A\u53ef\u5bf9\u89d2\u5316\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662fA\u6709n\u4e2a\u7ebf\u6027\u65e0\u5173\u7684\u7279\u5f81\u5411\u91cf\u3002<br \/>\n  \u4e8b\u5b9e\u4e0a\uff0c<span class=\"katex math inline\">A=PDP^{-1}<\/span>\uff0cD\u4e3a\u5bf9\u89d2\u77e9\u9635\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662fP\u7684\u5217\u5411\u91cf\u662fA\u7684n\u4e2a\u7ebf\u6027\u65e0\u5173\u7684\u7279\u5f81\u5411\u91cf\u3002\u6b64\u65f6\uff0cD\u7684\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u5206\u522b\u662fA\u7684\u5bf9\u5e94\u4e8eP\u4e2d\u7279\u5f81\u5411\u91cf\u7684\u7279\u5f81\u503c\u3002\n<\/p><\/blockquote>\n<p>\u6362\u53e5\u8bdd\u8bf4\uff0cA\u53ef\u5bf9\u89d2\u5316\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662f\u6709\u8db3\u591f\u7684\u7279\u5f81\u5411\u91cf\u5f62\u6210<span class=\"katex math inline\">R^n<\/span>\u7684\u57fa\uff0c\u6211\u4eec\u79f0\u8fd9\u6837\u7684\u57fa\u4e3a<strong>\u7279\u5f81\u5411\u91cf\u57fa<\/strong><\/p>\n<h1><span id=\"i\">\u77e9\u9635\u7684\u5bf9\u89d2\u5316<\/span><\/h1>\n<p><strong>\u4f8b3<\/strong> \u53ef\u80fd\u7684\u8bdd\uff0c\u5c06\u4e0b\u9762\u77e9\u9635\u5bf9\u89d2\u5316<\/p>\n<div class=\"katex math multi-line no-emojify\">A=\\left[\\begin{array}{rrr} 1&amp;3&amp;3 \\\\ -3&amp;-5&amp;-3 \\\\3&amp;3&amp;1\\\\ \\end{array}\\right]\n<\/div>\n<p>\u5373\u6c42\u53ef\u9006\u77e9\u9635P\u548c\u5bf9\u89d2\u77e9\u9635D\uff0c\u4f7f\u5f97<span class=\"katex math inline\">A=PDP^{-1}<\/span><br \/>\n<strong>\u89e3<\/strong>\u3000\u5bf9\u89d2\u5316\u5de5\u4f5c\u53ef\u5206\u4e3a4\u6b65\u6765\u5b8c\u6210\u3002<br \/>\n<strong>\u7b2c\u4e00\u6b65<\/strong>\u3000<strong>\u6c42\u51faA\u7684\u7279\u5f81\u503c<\/strong>\u3002\u57285.2\u8282\u66fe\u63d0\u5230\u8fc7\uff0c\u5728\u77e9\u9635\u7684\u89c4\u6a21\u5927\u4e8e2&#215;2\u65f6\uff0c\u53ef\u501f\u7528\u8ba1\u7b97\u673a\u6c42\u7279\u5f81\u503c\uff0c\u4e3a\u907f\u514d\u5206\u5fc3\uff0c\u672c\u4e66\u5c06\u4f1a\u63d0\u4f9b\u8fd9\u4e00\u6b65\u7684\u5185\u5bb9\uff0c\u73b0\u5728\u7684\u7279\u5f81\u65b9\u7a0b\u662f\u4e00\u4e2a3\u6b21\u591a\u9879\u5f0f\uff0c\u53ef\u5206\u89e3\u4e3a\uff1a<\/p>\n<div class=\"katex math multi-line no-emojify\">0=\\det (A-\\lambda I)=-\\lambda^3-3\\lambda^2+4=-(\\lambda-1)(\\lambda+2)^2\n<\/div>\n<p>\u7279\u5f81\u503c\u662f<span class=\"katex math inline\">\\lambda=1\u548c\\lambda=-2<\/span>.<br \/>\n<strong>\u7b2c\u4e8c\u6b65<\/strong>\u3000<strong>\u6c42A\u76843\u4e2a\u7ebf\u6027\u65e0\u5173\u7684\u7279\u5f81\u5411\u91cf<\/strong>\u3002\u56e0\u4e3aA\u662f3&#215;3\u7684\uff0c\u6545\u9700\u89813\u4e2a\u5411\u91cf\u3002\u8fd9\u4e00\u6b65\u5f88\u5173\u952e\uff0c\u5982\u679c\u627e\u4e0d\u5230\u8fd93\u4e2a\u5411\u91cf\uff0c\u90a3\u4e48\u7531\u5b9a\u74065\uff0cA\u5c31\u4e0d\u80fd\u5bf9\u89d2\u5316\uff0c\u75285.1\u8282\u7684\u65b9\u6cd5\u53ef\u6c42\u51fa\u6bcf\u4e00\u7279\u5f81\u7a7a\u95f4\u7684\u57fa\uff1a<\/p>\n<div class=\"katex math multi-line no-emojify\">\u5bf9\u4e8e\\lambda=1\u7684\u57fa\u662f\uff1av_1=\\left[\\begin{array}{rrr}1\\\\-1\\\\1\\end{array}\\right]\n<\/div>\n<div class=\"katex math multi-line no-emojify\">\u5bf9\u4e8e\\lambda=1\u7684\u57fa\u662f\uff1av_1=\\left[\\begin{array}{rrr}1\\\\-1\\\\1\\end{array}\\right]\n<\/div>\n<p>\u4f60\u53ef\u4ee5\u9a8c\u8bc1<span class=\"katex math inline\">&#92;{v_1,v_2,v_3&#92;}<\/span>\u662f\u7ebf\u6027\u65e0\u5173\u7684\u3002<br \/>\n<strong>\u7b2c3\u6b65<\/strong>\u3000<strong>\u7528\u7b2c2\u6b65\u5f97\u5230\u7684\u5411\u91cf\u6784\u9020\u77e9\u9635P<\/strong>\u5411\u91cf\u7684\u6b21\u5e8f\u4e0d\u91cd\u8981\uff0c\u7528\u7b2c\u4e8c\u6b65\u9009\u62e9\u7684\u6b21\u5e8f\uff0c\u5f62\u6210<\/p>\n<div class=\"katex math multi-line no-emojify\">p=\\begin{bmatrix}v_1&amp;v_2&amp;v_3\\end{bmatrix}=\\left[\\begin{array}{rrr}1&amp;-1&amp;-1\\\\-1&amp;1&amp;0\\\\1&amp;0&amp;1\\\\ \\end{array}\\right]\n<\/div>\n<p><strong>\u7b2c4\u6b65<\/strong>\u3000<strong>\u7528\u5bf9\u5e94\u7684\u7279\u5f81\u503c\u6784\u9020\u77e9\u9635D<\/strong>\u3002\u6784\u9020D\u65f6\uff0c\u7279\u5f81\u503c\u7684\u6b21\u5e8f\u5fc5\u987b\u548c\u77e9\u9635P\u9009\u62e9\u7684\u7279\u5f81\u5411\u91cf\u7684\u6b21\u5e8f\u4e00\u81f4\u3002\u5bf9\u5e94<span class=\"katex math inline\">\\lambda=-2<\/span>\u7684\u7279\u5f81\u5411\u91cf\u67092\u4e2a\uff0c\u7279\u5f81\u503c<span class=\"katex math inline\">\\lambda=-2<\/span>\u8981\u51fa\u73b0\u4e24\u6b21\uff1a<\/p>\n<div class=\"katex math multi-line no-emojify\">D=\\left[\\begin{array}{rrr}1&amp;0&amp;0\\\\0&amp;-2&amp;0\\\\0&amp;0&amp;-2\\\\ \\end{array}\\right]\n<\/div>\n<p>\u9a8c\u8bc1\u6240\u62db\u7684P\u548cD\u662f\u5426\u6b63\u597d\u662f\u4e00\u4e2a\u597d\u7684\u4e60\u60ef\uff0c\u4e3a\u907f\u514d\u8ba1\u7b97<span class=\"katex math inline\">P^{-1}<\/span>,\u53ef\u7b80\u5355\u9a8c\u8bc1<span class=\"katex math inline\">AP=PD<\/span>\uff0c\u8fd9\u7b49\u4ef7\u4e8e\u5f53P\u53ef\u9006\u65f6<span class=\"katex math inline\">A=PDP^{-1}<\/span>\u3002\uff08\u4f46\u5fc5\u987b\u786e\u8ba4P\u662f\u53ef\u9006\u7684\uff01\uff09\u6211\u4eec\u8ba1\u7b97<\/p>\n<div class=\"katex math multi-line no-emojify\">AP=\\left[\\begin{array}{rrr}1&amp;3&amp;3\\\\-3&amp;-5&amp;-3\\\\3&amp;3&amp;1\\\\ \\end{array}\\right]<br \/>\n\\left[\\begin{array}{rrr}1&amp;-1&amp;-1\\\\-1&amp;1&amp;0\\\\1&amp;0&amp;1\\\\ \\end{array}\\right]<br \/>\n=\\left[\\begin{array}{rrr}1&amp;2&amp;2\\\\-1&amp;-2&amp;0\\\\1&amp;0&amp;-2\\\\ \\end{array}\\right]\n<\/div>\n<div class=\"katex math multi-line no-emojify\">PD=\\left[\\begin{array}{rrr}1&amp;-1&amp;-1\\\\-1&amp;1&amp;\\0\\\\1&amp;0&amp;1\\\\ \\end{array}\\right]<br \/>\n\\left[\\begin{array}{rrr}1&amp;0&amp;0\\\\0&amp;-2&amp;0\\\\0&amp;0&amp;-2\\\\ \\end{array}\\right]<br \/>\n=\\left[\\begin{array}{rrr}1&amp;2&amp;2\\\\-1&amp;-2&amp;0\\\\1&amp;0&amp;-2\\\\ \\end{array}\\right]\n<\/div>\n<blockquote><p>\n  <strong>\u5b9a\u74066<\/strong>\u3000\u6709n\u4e2a\u76f8\u5f02\u7279\u5f81\u503c\u7684nxn\u77e9\u9635\u53ef\u5bf9\u89d2\u5316\u3002\n<\/p><\/blockquote>\n<p>\u4e0d\u8fc7\uff0cnxn\u77e9\u9635\u5e76\u4e0d\u9700\u8981n\u4e2a\u76f8\u5f02\u7279\u5f81\u503c\u624d\u53ef\u5bf9\u89d2\u5316\uff0c\u4f8b3\u76843&#215;3\u77e9\u9635\u5c3d\u7ba1\u53ea\u67092\u4e2a\u76f8\u5f02\u7684\u7279\u5f81\u503c\uff0c\u4f46\u5b83\u662f\u53ef\u5bf9\u89d2\u5316\u7684\u3002<\/p>\n<h1><span id=\"i-2\">\u7279\u5f81\u503c\u4e0d\u662f\u90fd\u76f8\u5f02\u7684\u77e9\u9635<\/span><\/h1>\n<p>\u5982\u679cnxn\u77e9\u9635A\u6709n\u4e2a\u76f8\u5f02\u7684\u7279\u5f81\u503c\u53ca\u76f8\u5e94\u7684\u7279\u5f81\u5411\u91cf<span class=\"katex math inline\">v_1,\\dots,v_n<\/span>,\u5982\u679c\u8bb0<span class=\"katex math inline\">P=\\begin{bmatrix}v_1&amp;\\dots &amp;v_n\\end{bmatrix}<\/span>\uff0c\u90a3\u4e48\u7531\u5b9a\u74062\uff0cP\u7684\u5217\u65f6\u7ebf\u6027\u65e0\u5173\u7684\uff0c\u81ea\u7136P\u662f\u53ef\u9006\u7684\u3002\u5f53A\u53ef\u5bf9\u89d2\u5316\uff0c\u4f46A\u76f8\u5f02\u7684\u7279\u5f81\u503c\u7684\u4e2a\u6570\u5c11\u4e8en\u65f6\uff0c\u6211\u4eec\u4ecd\u53ef\u4ee5\u7528\u4ee5\u4e0b\u5b9a\u7406\u7ed9\u51fa\u7684\u65b9\u6cd5\u6765\u6784\u9020\u53ef\u9006\u77e9\u9635P\u3002<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u74067<\/strong>\u3000\u8bbeA\u662fnxn\u77e9\u9635\uff0c\u5176\u76f8\u5f02\u7684\u7279\u5f81\u503c\u662f<span class=\"katex math inline\">\\lambda_1,\\dots,\\lambda_P<\/span>.<br \/>\n  a.\u3000\u5bf9\u4e8e<span class=\"katex math inline\">1\\le k\\le p<\/span>,<span class=\"katex math inline\">\\lambda_k<\/span>\u7684\u7279\u5f81\u7a7a\u95f4\u7684\u7ef4\u6570\u5c0f\u4e8e\u6216\u7b49\u4e8e<span class=\"katex math inline\">\\lambda_k<\/span>\u7684\u4ee3\u6570\u91cd\u6570\u3002<br \/>\n  b.\u3000\u77e9\u9635A\u53ef\u5bf9\u89d2\u5316\u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662f\u6240\u6709\u4e0d\u540c\u77e9\u9635\u7a7a\u95f4\u7684\u7ef4\u6570\u4e4b\u548c\u4e3an.\u5373\u6bcf\u4e2a<span class=\"katex math inline\">\\lambda_k<\/span>\u7684\u7279\u5f81\u7a7a\u95f4\u7684\u7ef4\u6570\u7b49\u4e8e<span class=\"katex math inline\">\\lambda_k<\/span>\u7684\u4ee3\u6570\u91cd\u6570\u3002<br \/>\n  c.\u3000\u82e5A\u53ef\u5bf9\u89d2\u5316\uff0c<span class=\"katex math inline\">\\beta_k<\/span>\u662f\u5bf9\u5e94\u4e8e<span class=\"katex math inline\">\\lambda_k<\/span>\u7684\u7279\u5f81\u7a7a\u95f4\u7684\u57fa\uff0c\u90a3\u4e48\uff0c\u96c6\u5408<span class=\"katex math inline\">\\beta_1,\\dots,\\beta_p<\/span>\u4e2d\u6240\u6709\u5411\u91cf\u7684\u96c6\u5408\u662f<span class=\"katex math inline\">R^n<\/span>\u7684\u7279\u5f81\u5411\u91cf\u57fa\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>Contents1 \u77e9\u9635\u7684\u5bf9\u89d2\u53162 \u7279\u5f81\u503c\u4e0d\u662f\u90fd\u76f8\u5f02\u7684\u77e9\u9635 \u5728\u5f88\u591a\u60c5\u51b5\u4e0b\uff0c\u4ece\u5206\u89e3\u5f0fA=PDP^{-1},\u6211\u4eec\u80fd [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":410,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,22,26],"tags":[],"class_list":["post-549","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-21","category-22","category-26"],"_links":{"self":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/549","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=549"}],"version-history":[{"count":3,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/549\/revisions"}],"predecessor-version":[{"id":552,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/549\/revisions\/552"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/media\/410"}],"wp:attachment":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=549"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=549"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=549"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}