{"id":585,"date":"2021-05-24T19:55:18","date_gmt":"2021-05-24T11:55:18","guid":{"rendered":"http:\/\/www.wayln.com\/?p=585"},"modified":"2021-05-24T20:13:00","modified_gmt":"2021-05-24T12:13:00","slug":"5-5-%e5%a4%8d%e7%89%b9%e5%be%81%e5%80%bc","status":"publish","type":"post","link":"https:\/\/www.wayln.com\/?p=585","title":{"rendered":"5.5 \u590d\u7279\u5f81\u503c"},"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> \u5411\u91cf\u7684\u5b9e\u90e8\u4e0e\u865a\u90e8<\/a><\/li><li><a href=\"#Cn\"><span class=\"toc_number toc_depth_1\">2<\/span> \u4f5c\u7528\u4e8eC^n\u4e0a\u7684\u5b9e\u77e9\u9635\u7684\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf<\/a><\/li><\/ul><\/div>\n<p>nxn\u77e9\u9635\u7684\u7279\u5f81\u65b9\u7a0b\u542b\u6709n\u6b21\u591a\u9879\u5f0f\uff0c\u5982\u679c\u8003\u8651\u590d\u6839\uff0c\u65b9\u7a0b\u6070\u597d\u6709n\u4e2a\u6839\uff0c\u91cd\u6839\u91cd\u590d\u8ba1\u7b97\u3002\u8ba9A\u4f5c\u7528\u4e8en\u7ef4\u590d\u7a7a\u95f4<span class=\"katex math inline\">C^n<\/span>\u4e2d\u7684n\u5143\u590d\u6570\u3002<br \/>\n\u5bf9\u590d\u7279\u5f81\u503c\u7684\u7814\u7a76\u80fd\u4f7f\u6211\u4eec\u53bb\u63ed\u793a\u5404\u79cd\u5b9e\u9645\u751f\u6d3b\u95ee\u9898\u4e2d\u51fa\u73b0\u7684\u67d0\u4e9b\u5b9e\u77e9\u9635\u4e2d\u9690\u85cf\u7684\u4fe1\u606f\uff0c\u8fd9\u4e9b\u95ee\u9898\u5305\u62ec\u5f88\u591a\u8574\u6db5\u5468\u671f\u8fd0\u52a8\u7684\u5b9e\u52a8\u529b\u7cfb\u7edf\u3001\u632f\u52a8\u6216\u7a7a\u95f4\u7684\u67d0\u79cd\u65cb\u8f6c\u3002<br \/>\n\u4e00\u4e2a\u590d\u6570<span class=\"katex math inline\">\\lambda<\/span>\u6ee1\u8db3<span class=\"katex math inline\">\\det(A-\\lambda I)=0<\/span>\u5f53\u4e14\u4ec5\u5f53\u5728<span class=\"katex math inline\">C^n<\/span>\u4e2d\u5b58\u5728\u4e00\u4e2a\u975e\u96f6\u5411\u91cf<span class=\"katex math inline\">x<\/span>\uff0c\u4f7f\u5f97<span class=\"katex math inline\">Ax=\\lambda x<\/span>\u3002\u6211\u4eec\u79f0\u8fd9\u6837\u7684<span class=\"katex math inline\">\\lambda<\/span>\u662f\u590d\u7279\u5f81\u503c\uff0cx\u662f\u5bf9\u5e94<span class=\"katex math inline\">\\lambda<\/span>\u7684\uff08\u590d\uff09\u7279\u5f81\u5411\u91cf\u3002<\/p>\n<blockquote><p>\n  <strong>\u4f8b 2<\/strong> \u8bbe<span class=\"katex math inline\">A=\\left[\\begin{array}{rr}0.5&amp;-0.6&#92;\\0.75&amp;1.1\\end{array}\\right]<\/span>,\u6c42A\u7684\u7279>\u5f81\u503c\u53ca\u6bcf\u4e2a\u7279\u5f81\u7a7a\u95f4\u7684\u57fa\u3002<br \/>\n  <strong>\u89e3<\/strong> A\u7684\u7279\u5f81\u65b9\u7a0b\u662f<\/p>\n<div class=\"katex math multi-line no-emojify\">0=\\det \\begin{bmatrix}0.5-\\lambda &amp; -0.6\\\\0.75&amp;1.1-\\lambda\\end{bmatrix}\\\\<br \/>\n  =(0.5-\\lambda)(1.1-\\lambda)-(-0.6)(0.75)\\\\<br \/>\n  =\\lambda^2-1.6\\lambda+1\n  <\/div>\n<p>  \u4f9d\u6c42\u6839\u516c\u5f0f\uff0c\u5f97<span class=\"katex math inline\">\\lambda=\\frac{1}{2}\\left[1.6 \\pm \\sqrt{(-1.6)^2-4} \\right]=0.8 \\pm 0.6i<\/span>\u3002\u5bf9\u7279\u5f81\u503c<span class=\"katex math inline\">\\lambda=0.8-0.6i<\/span>,\u6784\u9020<\/p>\n<div class=\"katex math multi-line no-emojify\">A-(0.8-0.6i)I=\\left[\\begin{array}{rr}0.5&amp;-0.6\\\\0.75&amp;1.1\\end{array}\\right]-\\begin{bmatrix}-0.3+0.6i&amp;-0.6\\\\0.75&amp;0.9+0.6i\\end{bmatrix} \\tag{1}\n  <\/div>\n<p>  \u7531\u4e8e\u6709\u590d\u6570\u8fd0\u7b97\uff0c\u624b\u5de5\u505a\u5bf9\u589e\u5149\u77e9\u9635\u884c\u5316\u7b80\u662f\u76f8\u5f53\u6162\u7684\u3002\u4e0d\u8fc7\uff0c\u7ec6\u5fc3\u89c2\u5bdf\u80fd\u627e\u5230\u5316\u7b80\u95ee\u9898\u7684\u65b9\u6cd5\uff1a\u56e0\u4e3a<span class=\"katex math inline\">0.8-0.6i<\/span>\u662f\u7279\u5f81\u503c\uff0c\u6211\u4eec\u77e5\u9053\u65b9\u7a0b\u7ec4\uff1a<\/p>\n<div class=\"katex math multi-line no-emojify\">\\begin{cases}<br \/>\n  (-0.3+0.6i)x_1-0.6x_2=0\\\\<br \/>\n  0.75x_1+(0.3+0.6i)x_2=0 \\tag{2}<br \/>\n  \\end{cases}\n  <\/div>\n<p>  \u6709\u975e\u96f6\u89e3\uff08\u8fd9\u91cc\u7684<span class=\"katex math inline\">x_1\u548cx_2<\/span>\u53ef\u80fd\u662f\u590d\u6570\uff09\uff0c\u56e0\u6b64(2)\u4e2d\u7684\u4e24\u4e2a\u65b9\u7a0b\u786e\u5b9a\u7684<span class=\"katex math inline\">x_1\u4e0ex_2<\/span>\u4e4b\u95f4\u7684\u5173\u7cfb\u65f6\u7edf\u4e00\u5173\u7cfb\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u901a\u8fc7\u5176\u4e2d\u7684\u4e00\u4e2a\u65b9\u7a0b\u5c06\u67d0\u4e2a\u53d8\u91cf\u7528\u53e6\u4e00\u4e2a\u6765\u8868\u793a\u3002<br \/>\n  \u7531(2)\u7684\u7b2c\u4e8c\u4e2a\u65b9\u7a0b\u5f97<\/p>\n<div class=\"katex math multi-line no-emojify\">0.75x_1=(-0.3-0.6i)x_2\\\\<br \/>\n  x_1=(-0.4-0.8i)x_2\n  <\/div>\n<p>  \u4e3a\u53bb\u6389\u5c0f\u6570\uff0c\u53d6<span class=\"katex math inline\">x_2=5<\/span>,\u6709<span class=\"katex math inline\">x_1=-2-4i<\/span>.\u5bf9\u5e94<span class=\"katex math inline\">\\lambda=0.8-0.6i<\/span>\u7684\u7279\u5f81\u7a7a\u95f4\u7684\u57fa\u662f<\/p>\n<div class=\"katex math multi-line no-emojify\">v_1=\\begin{bmatrix}-2-4i\\\\5\\end{bmatrix}\n  <\/div>\n<p>  \u5bf9<span class=\"katex math inline\">\\lambda=0.8+0.6i<\/span>\u505a\u540c\u6837\u7684\u8ba1\u7b97\uff0c\u53ef\u6c42\u51fa\u7279\u5f81\u5411\u91cf<\/p>\n<div class=\"katex math multi-line no-emojify\">v_2=\\begin{bmatrix}-2+4i\\\\5\\end{bmatrix}\n  <\/div>\n<p>  \u4e3a\u4e86\u9a8c\u8bc1\u7ed3\u679c\u662f\u5426\u6b63\u786e\uff0c\u53ef\u4ee5\u8ba1\u7b97<\/p>\n<div class=\"katex math multi-line no-emojify\">Av_2=\\left[\\begin{array}{rr}0.5&amp;-0.6\\\\0.75&amp;1.1\\end{array}\\right]\\begin{bmatrix}-2+4i\\\\5\\end{bmatrix}\\\\=\\begin{bmatrix}-4+2i\\\\4+3i\\end{bmatrix}=(0.8+0.6i)v_2\n  <\/div>\n<p>  \u975e\u5e38\u610f\u5916\uff0c\u4f8b2\u7684\u77e9\u9635A\u786e\u5b9a\u7684\u53d8\u6362<span class=\"katex math inline\">x\\mapsto Ax<\/span>\u5b9e\u9645\u4e0a\u662f\u4e00\u4e2a\u65cb\u8f6c\u53d8\u6362\u3002\u753b\u51fa\u4e00\u4e9b\u9002\u5f53\u7684\u70b9\u540e\uff0c\u8fd9\u4e00\u4e8b\u5b9e\u4f1a\u53d8\u5f97\u66f4\u660e\u663e\u3002\n<\/p><\/blockquote>\n<h1><span id=\"i\">\u5411\u91cf\u7684\u5b9e\u90e8\u4e0e\u865a\u90e8<\/span><\/h1>\n<p><span class=\"katex math inline\">C^n<\/span>\u7684\u590d\u5411\u91cfx\u7684\u5171\u8f6d\u5411\u91cf<span class=\"katex math inline\">\\bar{x}<\/span>\u4e5f\u662f<span class=\"katex math inline\">C^n<\/span>\u7684\u5411\u91cf\uff0c\u5b83\u7684\u5206\u91cf\u662fx\u5bf9\u5e94\u5206\u91cf\u7684\u5171\u8f6d\u590d\u6570\uff0c\u5411\u91cf<span class=\"katex math inline\">Re  x<\/span>\u548c<span class=\"katex math inline\">Imx<\/span>\u79f0\u4e3a\u590d\u5411\u91cfx\u7684\u5b9e\u90e8\u548c\u865a\u90e8\uff0c\u5206\u522b\u7531x\u7684\u5206\u91cf\u7684\u5b9e\u90e8\u548c\u865a\u90e8\u7ec4\u6210<br \/>\n\u5047\u8bbeB\u662f\u53ef\u80fd\u6709\u590d\u5143\u7d20\u7684mxn\u77e9\u9635\uff0c\u90a3\u4e48\uff0c\u4ee5B\u7684\u5143\u7d20\u7684\u5171\u8f6d\u590d\u6570\u4e3a\u5143\u7d20\u7684\u77e9\u9635\u8bb0\u4e3a<span class=\"katex math inline\">\\bar{B}<\/span>\u3002\u590d\u6570\u7684\u5171\u8f6d\u8fd0\u7b97\u6027\u8d28\u5bf9\u590d\u77e9\u9635\u4ee3\u6570\u4ea6\u6210\u7acb\uff1a<\/p>\n<div class=\"katex math multi-line no-emojify\">\\overline{rx}=\\bar{r}\\bar{x},\\overline{Bx}=\\bar{B}\\bar{x},\\overline{BC}=\\bar{b}\\bar{C},\\overline{rB}=\\bar{r}\\bar{B}\n<\/div>\n<h1><span id=\"Cn\">\u4f5c\u7528\u4e8eC^n\u4e0a\u7684\u5b9e\u77e9\u9635\u7684\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf<\/span><\/h1>\n<p>\u8bbeA\u4e3anxn\u7684\u5b9e\u77e9\u9635\uff0c\u90a3\u4e48<span class=\"katex math inline\">\\overline{Ax}=\\bar{A}\\bar{x}=A\\bar{x}<\/span>,\u5047\u5982<span class=\"katex math inline\">\\lambda<\/span>\u662fA\u7684\u7279\u5f81\u503c\uff0cx\u662f\u5bf9\u5e94<span class=\"katex math inline\">\\lambda<\/span>\u7684\u7279\u5f81\u5411\u91cf\uff0c\u90a3\u4e48<\/p>\n<div class=\"katex math multi-line no-emojify\">A\\bar{x}=\\overline{Ax}=\\overline{\\lambda x}=\\bar{\\lambda}\\bar{x}\n<\/div>\n<p>\u6545<span class=\"katex math inline\">\\bar{\\lambda}<\/span>\u540c\u6837\u662fA\u7684\u7279\u5f81\u503c\uff0c\u800c<span class=\"katex math inline\">\\bar{x}<\/span>\u662f\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u3002<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u7406 9<\/strong>\u3000\u8bbeA\u662f2&#215;2\u5b9e\u77e9\u9635\uff0c\u6709\u590d\u7279\u5f81\u503c<span class=\"katex math inline\">\\lambda=a-bi(b \\ne0)<\/span>\u53ca\u5bf9\u5e94\u7684<span class=\"katex math inline\">C^2<\/span>\u4e2d\u7684\u590d\u7279\u5f81\u5411\u91cfv\uff0c\u90a3\u4e48\n<\/p><\/blockquote>\n<div class=\"katex math multi-line no-emojify\">A=PCP^{-1},\u5176\u4e2dP=\\left[\\begin{array}{r} Rev &amp; Imv \\end{array} \\right],C=\\left[\\begin{array}{rr}a&amp;-b\\\\b&amp;a\\end{array}\\right]\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Contents1 \u5411\u91cf\u7684\u5b9e\u90e8\u4e0e\u865a\u90e82 \u4f5c\u7528\u4e8eC^n\u4e0a\u7684\u5b9e\u77e9\u9635\u7684\u7279\u5f81\u503c\u548c\u7279\u5f81\u5411\u91cf nxn\u77e9\u9635\u7684\u7279\u5f81\u65b9\u7a0b\u542b\u6709n\u6b21 [&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-585","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\/585","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=585"}],"version-history":[{"count":3,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/585\/revisions"}],"predecessor-version":[{"id":588,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/585\/revisions\/588"}],"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=585"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=585"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=585"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}