{"id":500,"date":"2021-05-11T21:42:07","date_gmt":"2021-05-11T13:42:07","guid":{"rendered":"http:\/\/www.wayln.com\/?p=500"},"modified":"2021-05-16T19:24:40","modified_gmt":"2021-05-16T11:24:40","slug":"5-2-%e7%89%b9%e5%be%81%e6%96%b9%e7%a8%8b","status":"publish","type":"post","link":"https:\/\/www.wayln.com\/?p=500","title":{"rendered":"5.2 \u7279\u5f81\u65b9\u7a0b"},"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> \u884c\u5217\u5f0f<\/a><\/li><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_1\">2<\/span> \u7279\u5f81\u65b9\u7a0b<\/a><\/li><li><a href=\"#i-3\"><span class=\"toc_number toc_depth_1\">3<\/span> \u5e94\u7528\u5230\u52a8\u529b\u7cfb\u7edf<\/a><\/li><\/ul><\/div>\n<h1><span id=\"i\">\u884c\u5217\u5f0f<\/span><\/h1>\n<p>\u3000\u8bbeA\u662fnxn\u77e9\u9635\uff0cU\u662f\u5bf9A\u4f5c\u884c\u66ff\u6362\u548c\u884c\u4ea4\u6362\uff08\u4e0d\u4f5c\u884c\u500d\u79f0\uff09\u6240\u5f97\u5230\u7684\u4efb\u4e00\u9636\u68af\u5f62\u77e9\u9635\uff0cr\u662f\u884c\u4ea4\u6362\u6b21\u6570\uff0c\u90a3\u4e48A\u7684 <strong>\u884c\u5217\u5f0f<\/strong> <span class=\"katex math inline\">\\det A=(-1)^ru_{11}\\dots u_{nn}<\/span>\u3002\u5982\u679cA\u53ef\u9006\uff0c\u90a3\u4e48<span class=\"katex math inline\">u_{11},\\dots,u_{nn}<\/span>\u90fd\u662f\u4e3b\u5143\u3002\u5426\u5219\uff0c\u81f3\u5c11\u6709<span class=\"katex math inline\">u_{nn}<\/span>\u4e3a\u96f6\uff0c\u4ece\u800c\u4e58\u79ef<span class=\"katex math inline\">u_{11}\\dots u_{nn}<\/span>\u4e3a\u96f6\u3002\u56e0\u6b64<\/p>\n<div class=\"katex math multi-line no-emojify\">\\det A=<br \/>\n \\begin{cases}<br \/>\n (-1)^ru_{11}\\dots u_{nn}\u3000 {\\text{\u5f53A\u53ef\u9006}}\\\\<br \/>\n 0\u3000 {\\text{\u5f53A\u4e0d\u53ef\u9006}}\u3000\\\\<br \/>\n \\end{cases} \\tag{1}\n<\/div>\n<blockquote><p>\n  <strong>\u5b9a\u7406<\/strong>\uff08\u53ef\u9006\u77e9\u9635\u5b9a\u7406\uff08\u7eed\uff09\uff09<br \/>\n  \u8bbeA\u662fnxn\u77e9\u9635\uff0c\u5219A\u662f\u53ef\u9006\u7684\u5f53\u4e14\u4ec5\u5f53<br \/>\n  s.\u30000\u4e0d\u662fA\u7684\u7279\u5f81\u503c\u3002<br \/>\n  t.\u3000A\u7684\u884c\u5217\u5f0f\u4e0d\u7b49\u4e8e\u96f6\u3002\n<\/p><\/blockquote>\n<p>\u5982\u679cA\u662f3&#215;3\u77e9\u9635\uff0c\u90a3\u4e48<span class=\"katex math inline\">\\left| \\det A \\right|<\/span>\u662f\u7531A\u7684\u5217\u5411\u91cf<span class=\"katex math inline\">a_1,a_2,a_3<\/span>\u751f\u6210\u7684\u5e73\u884c\u516d\u9762\u4f53\u7684\u4f53\u79ef\uff0c\u89c1\u4e0b\u56fe\u3002\u5e73\u884c\u516d\u9762\u4f53\u7684\u4f53\u79ef\u4e0d\u7b49\u4e8e\u96f6\u5f53\u4e14\u4ec5\u5f53\u5411\u91cf<span class=\"katex math inline\">a_1,a_2,a_3<\/span>\u7ebf\u6027\u65e0\u5173\uff0c\u5373\u77e9\u9635A\u662f\u53ef\u9006\u7684\u3002\uff08\u5982\u679c\u5411\u91cf\u975e\u96f6\u4e14\u7ebf\u6027\u76f8\u5173\uff0c\u4ed6\u4eec\u4f4d\u4e8e\u540c\u4e00\u4e2a\u5e73\u9762\u6216\u4e00\u6761\u76f4\u7ebf\u4e0a\uff09<br \/>\n<a class=\"wp-editor-md-post-content-link\" href=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/05\/wp_editor_md_387e89984d08e833a10a8d6c4bd9a2f7.jpg\"><img decoding=\"async\" src=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/05\/wp_editor_md_387e89984d08e833a10a8d6c4bd9a2f7.jpg\" alt=\"\" \/><\/a><\/p>\n<blockquote><p>\n  <strong>\u5b9a\u74063<\/strong> (\u884c\u5217\u5f0f\u7684\u6027\u8d28)<br \/>\n  \u8bbeA\u548cB\u662fnxn\u77e9\u9635<br \/>\n  a.\u3000A\u53ef\u9006\u7684\u5145\u8981\u6761\u4ef6\u662f<span class=\"katex math inline\">\\det A \\ne 0<\/span><br \/>\n  b.\u3000<span class=\"katex math inline\">\\det AB=(\\det A)(\\det B)<\/span><br \/>\n  c.\u3000<span class=\"katex math inline\">\\det A^T=\\det A<\/span><br \/>\n  d.\u3000\u82e5A\u662f\u4e09\u89d2\u5f62\u77e9\u9635\uff0c\u90a3\u4e48<span class=\"katex math inline\">\\det A<\/span>\u662fA\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20\u7684\u4e58\u79ef<br \/>\n  e.\u3000\u5bf9A\u4f5c\u884c\u66ff\u6362\u4e0d\u6539\u53d8\u5176\u884c\u5217\u5f0f\u503c\u3002\u505a\u4e00\u6b21\u822a\u4ea4\u6362\u4f7f\u5176\u884c\u5217\u5f0f\u503c\u7b26\u53f7\u6539\u53d8\u4e00\u6b21\u3002\u6570\u4e58\u4e00\u884c\u540e\uff0c\u884c\u5217\u5f0f\u503c\u7b49\u4e8e\u7528\u6b64\u6570\u4e58\u539f\u6765\u7684\u884c\u5217\u5f0f\u503c\u3002\n<\/p><\/blockquote>\n<h1><span id=\"i-2\">\u7279\u5f81\u65b9\u7a0b<\/span><\/h1>\n<p>\u5229\u7528\u5b9a\u74063\uff08a\uff09,\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u884c\u5217\u5f0f\u6765\u5224\u65ad\u77e9\u9635<span class=\"katex math inline\">A-\\lambda I<\/span>\u662f\u5426\u53ef\u9006\u3002\u6570\u503c\u65b9\u7a0b<span class=\"katex math inline\">\\det (A-\\lambda I)=0<\/span>\u79f0\u4e3aA\u7684<strong>\u7279\u5f81\u65b9\u7a0b<\/strong>\u3002<\/p>\n<blockquote><p>\n  \u6570<span class=\"katex math inline\">\\lambda<\/span>\u662fnxn\u77e9\u9635A\u7684\u7279\u5f81\u503c\u7684\u5145\u8981\u6761\u4ef6\u662f<span class=\"katex math inline\">\\lambda<\/span>\u662f\u7279\u5f81\u65b9\u7a0b<span class=\"katex math inline\">\\det (A-\\lambda I)=0<\/span>\u7684\u6839\u3002\n<\/p><\/blockquote>\n<p><strong>\u4f8b3<\/strong> \u6c42<span class=\"katex math inline\">A=\\left[ \\begin{array}{rrrr}5&amp;-2&amp;6&amp;-1&#92;\\0&amp;3&amp;-8&amp;0&#92;\\0&amp;0&amp;5&amp;4&#92;\\0&amp;0&amp;0&amp;1&#92;\\  \\end{array}\\right]<\/span>\u7684\u7279\u5f81\u65b9\u7a0b<br \/>\n<strong>\u89e3<\/strong> \u5199\u51fa<span class=\"katex math inline\">A-\\lambda I<\/span>\u5e76\u5229\u7528\u5b9a\u74063\uff08d\uff09\uff1a<\/p>\n<div class=\"katex math multi-line no-emojify\">det(A-\\lambda I)=\\det \\left[\\begin{array}{cccc}5-\\lambda&amp;-2&amp;6&amp;-1\\\\<br \/>\n0&amp;3-\\lambda&amp;-8&amp;0\\\\<br \/>\n0&amp;0&amp;5-\\lambda&amp;4\\\\<br \/>\n0&amp;0&amp;0&amp;1-\\lambda\\\\<br \/>\n\\end{array}\\right]\n<\/div>\n<div class=\"katex math multi-line no-emojify\">=(5-\\lambda)(3-\\lambda)(5-\\lambda)(1-\\lambda)\n<\/div>\n<p>\u7279\u5f81\u65b9\u7a0b\u662f<\/p>\n<div class=\"katex math multi-line no-emojify\">(5-\\lambda)^2(3-\\lambda)(1-\\lambda)=0\n<\/div>\n<p>\u6216<\/p>\n<div class=\"katex math multi-line no-emojify\">(5-\\lambda)^2(\\lambda-3)(\\lambda-1)=0\n<\/div>\n<p>\u5c55\u5f00\u4e58\u79ef\uff0c\u7279\u5f81\u65b9\u7a0b\u4e5f\u53ef\u4ee5\u5199\u4e3a<\/p>\n<div class=\"katex math multi-line no-emojify\">\\lambda^4-14\\lambda^3+68\\lambda^2-130\\lambda+75=0\n<\/div>\n<p>\u4f8b1\u548c\u4f8b3\u7684<span class=\"katex math inline\">\\det(A-\\lambda I)<\/span>\u662f\u5173\u4e8e<span class=\"katex math inline\">\\lambda<\/span>\u7684\u591a\u9879\u5f0f\uff0c\u53ef\u4ee5\u770b\u51fa\uff0c\u5982\u679cA\u662fnxn\u77e9\u9635\uff0c\u90a3\u4e48<span class=\"katex math inline\">\\det(A-\\lambda I)<\/span>\u662fn\u6b21\u591a\u9879\u5f0f\uff0c\u79f0\u4e3aA\u7684\u7279\u5f81\u591a\u9879\u5f0f<\/p>\n<p>\u5728\u4f8b3\u4e2d\uff0c\u7531\u4e8e\u56e0\u5b50<span class=\"katex math inline\">(\\lambda-5)<\/span>\u5728\u7279\u5f81\u591a\u9879\u5f0f\u4e2d\u51fa\u73b0\u4e862\u6b21\uff0c\u6545\u79f0\u7279\u5f81\u662f5\u6709\u91cd\u65702,\u3002\u4e00\u822c\u5730\uff0c\u628a\u7279\u5f81\u503c<span class=\"katex math inline\">\\lambda<\/span>\u4f5c\u4e3a\u7279\u5f81\u65b9\u7a0b\u6839\u7684\u91cd\u6570\u79f0\u4e3a\u662f<span class=\"katex math inline\">\\lambda<\/span>\u7684<strong>\uff08\u4ee3\u6570\uff09\u91cd\u6570<\/strong>\u3002<\/p>\n<p>\u56e0\u4e3anxn\u77e9\u9635\u7684\u7279\u5f81\u65b9\u7a0b\u5305\u542b\u6709\u4e00\u4e2an\u6b21\u591a\u9879\u5f0f\uff0c\u5982\u679c\u7b97\u4e0a\u91cd\u6839\uff0c\u5e76\u5141\u8bb8\u6709\u590d\u6839\uff0c\u5219\u7279\u5f81\u65b9\u7a0b\u6070\u597d\u6709n\u4e2a\u6839\u3002\u590d\u6839\u79f0\u4e3a<strong>\u590d\u7279\u5f81\u503c<\/strong><\/p>\n<p>#\u76f8\u4f3c\u6027<br \/>\n\u5047\u5982A\u548cB\u662fnxn\u77e9\u9635\uff0c\u5982\u679c\u5b58\u5728\u53ef\u9006\u77e9\u9635P\uff0c\u4f7f\u5f97<span class=\"katex math inline\">P^{-1}AP=B<\/span>,\u6216\u7b49\u4ef7\u5730<span class=\"katex math inline\">A=PBP^{-1}<\/span>\uff0c\u5219\u79f0<strong>A\u76f8\u4f3c\u4e8eB<\/strong>\u3002\u8bb0<span class=\"katex math inline\">Q=P^{-1}<\/span>\uff0c\u5219\u6709<span class=\"katex math inline\">Q^{-1}BQ=A<\/span>\uff0c\u5373B\u4e5f\u76f8\u4f3c\u4e8eA\u3002\u6545\u6211\u4eec\u7b80\u5355\u8bf4A\u548cB\u662f\u76f8\u4f3c\u7684\u3002\u628aA\u53d8\u6210<span class=\"katex math inline\">P^{-1}AP<\/span>\u7684\u53d8\u6362\u79f0\u4e3a<strong>\u76f8\u4f3c\u53d8\u6362<\/strong>\u3002<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u74064<\/strong>\u3000\u82e5nxn\u77e9\u9635A\u548cB\u662f\u76f8\u4f3c\u7684\uff0c\u90a3\u4e48\u4ed6\u4eec\u6709\u76f8\u540c\u7684\u7279\u5f81\u591a\u9879\u5f0f\uff0c\u4ece\u800c\u6709\u76f8\u540c\u7684\u7279\u5f81\u503c\uff08\u548c\u91cd\u6570\uff09<\/p>\n<p>  <strong>\u8b66\u544a<\/strong><br \/>\n  1.\u3000\u77e9\u9635<span class=\"katex math inline\">\\begin{bmatrix}2&amp;1&#92;\\0&amp;2\\end{bmatrix}<\/span>\u548c><span class=\"katex math inline\">\\begin{bmatrix}1&amp;0&#92;\\0&amp;2\\end{bmatrix}<\/span>\u5373\u4f7f\u6709\u76f8\u540c\u7684\u7279\u5f81\u503c\u4e5f\u4e0d\u76f8\u4f3c\u3002<br \/>\n  2.\u3000\u76f8\u4f3c\u6027\u4e0e\u884c\u7b49\u4ef7\u4e0d\u662f\u4e00\u56de\u4e8b\uff08\u52a0\u5165A\u884c\u7b49\u4ef7\u4e0eB\uff0c\u5219\u5b58\u5728\u53ef\u9006\u77e9\u9635E,\u4f7f\u5f97<span class=\"katex math inline\">1B=EA<\/span>\uff09\u5bf9\u77e9\u9635\u4f5c\u884c\u53d8\u6362\u901a\u5e38\u4f1a\u6539\u53d8\u77e9\u9635\u7684\u7279\u5f81\u503c\n<\/p><\/blockquote>\n<h1><span id=\"i-3\">\u5e94\u7528\u5230\u52a8\u529b\u7cfb\u7edf<\/span><\/h1>\n","protected":false},"excerpt":{"rendered":"<p>Contents1 \u884c\u5217\u5f0f2 \u7279\u5f81\u65b9\u7a0b3 \u5e94\u7528\u5230\u52a8\u529b\u7cfb\u7edf \u884c\u5217\u5f0f \u3000\u8bbeA\u662fnxn\u77e9\u9635\uff0cU\u662f\u5bf9A\u4f5c\u884c\u66ff\u6362\u548c\u884c\u4ea4\u6362 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":388,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,22,26],"tags":[],"class_list":["post-500","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\/500","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=500"}],"version-history":[{"count":24,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/500\/revisions"}],"predecessor-version":[{"id":548,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/500\/revisions\/548"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/media\/388"}],"wp:attachment":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=500"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=500"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=500"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}