{"id":367,"date":"2021-04-15T18:45:24","date_gmt":"2021-04-15T10:45:24","guid":{"rendered":"http:\/\/www.wayln.com\/?p=367"},"modified":"2021-04-15T21:11:21","modified_gmt":"2021-04-15T13:11:21","slug":"3-3-%e5%85%8b%e6%8b%89%e9%bb%98%e6%b3%95%e5%88%99%e3%80%81%e4%bd%93%e7%a7%af%e5%92%8c%e7%ba%bf%e6%80%a7%e5%8f%98%e6%8d%a2","status":"publish","type":"post","link":"https:\/\/www.wayln.com\/?p=367","title":{"rendered":"3.3 \u514b\u62c9\u9ed8\u6cd5\u5219\u3001\u4f53\u79ef\u548c\u7ebf\u6027\u53d8\u6362"},"content":{"rendered":"<div id=\"toc_container\" class=\"toc_transparent no_bullets\"><p class=\"toc_title\">Contents<\/p><ul class=\"toc_list\"><li><a href=\"#A-1\"><span class=\"toc_number toc_depth_1\">1<\/span> \u4e00\u4e2a\u6c42A^{-1}\u7684\u516c\u5f0f<\/a><\/li><li><a href=\"#i\"><span class=\"toc_number toc_depth_1\">2<\/span> \u7ebf\u6027\u53d8\u6362<\/a><\/li><\/ul><\/div>\n<p>\u514b\u62c9\u9ed8\u6cd5\u5219\u5728\u5404\u79cd\u7406\u8bba\u8ba1\u7b97\u4e2d\u662f\u5fc5\u9700\u7684\uff0c\u4f8b\u5982\uff0c\u5b83\u88ab\u7528\u6765\u7814\u7a76<span class=\"katex math inline\">Ax=b<\/span>\u7684\u89e3\u53d7b\u4e2d\u6570\u503c\u7684\u53d8\u5316\u800c\u53d7\u5230\u4ec0\u4e48\u6837\u7684\u5f71\u54cd\u3002\u7136\u800c\u8fd9\u4e2a\u516c\u5f0f\u5bf9\u624b\u7b97\u662f\u6ca1\u6709\u591a\u5927\u6548\u679c\u7684\uff0c\u9664\u975e\u662f2&#215;2\u62163&#215;3\u77e9\u9635\u3002<br \/>\n\u5bf9\u4efb\u610fnxn\u77e9\u9635A\u548c\u4efb\u610f\u7684<span class=\"katex math inline\">R^n<\/span>\u4e2d\u5411\u91cfb,\u4ee4<span class=\"katex math inline\">A_i(b)<\/span>\u8868\u793aA\u4e2d\u7b2ci\u5217\u7531\u5411\u91cfb\u66ff\u6362\u5f97\u5230\u7684\u77e9\u9635<\/p>\n<div class=\"katex math multi-line no-emojify\">A_i(b)=\\begin{bmatrix}<br \/>\na_1&amp;\\dots&amp;\\underbrace{b}_{\u7b2ci\u5217}&amp;\\dots&amp;a_n<br \/>\n\\end{bmatrix}\n<\/div>\n<blockquote><p>\n  <strong>\u5b9a\u74067<\/strong>\u3000\u514b\u62c9\u9ed8\u6cd5\u5219<br \/>\n  \u8bbeA\u662f\u4e00\u4e2a\u53ef\u9006\u7684nxn\u77e9\u9635\uff0c\u5bf9<span class=\"katex math inline\">R^n<\/span>\u4e2d\u4efb\u610f\u5411\u91cfb,\u65b9\u7a0b<span class=\"katex math inline\">Ax=b<\/span>\u7684\u552f\u4e00\u89e3\u53ef\u7531\u4e0b\u5f0f\u7ed9\u51fa<\/p>\n<div class=\"katex math multi-line no-emojify\">x_i=\\frac{\\det A_i(b)}{\\det A},i=1,2,\\dots,n  \\tag{1}\n  <\/div>\n<\/blockquote>\n<h1><span id=\"A-1\">\u4e00\u4e2a\u6c42A^{-1}\u7684\u516c\u5f0f<\/span><\/h1>\n<p>\u514b\u62c9\u9ed8\u6cd5\u5219\u53ef\u4ee5\u5bb9\u6613\u7684\u5bfc\u51fa\u4e00\u4e2a\u6c42nxn\u77e9\u9635A\u7684\u9006\u7684\u4e00\u822c\u516c\u5f0f\u3002<span class=\"katex math inline\">A^{-1}<\/span>\u7684\u7b2cj\u5217\u65f6\u4e00\u4e2a\u5411\u91cfx,\u6ee1\u8db3<\/p>\n<div class=\"katex math multi-line no-emojify\">Ax=e_j\n<\/div>\n<p>\u6b64\u5904<span class=\"katex math inline\">e_j<\/span>\u662f\u5355\u4f4d\u77e9\u9635\u7684\u7b2cj\u5217\uff0cx\u7684\u7b2ci\u4e2a\u6570\u503c\u662f<span class=\"katex math inline\">A^{-1}<\/span>\u4e2d(i\uff0cj)\u4f4d\u7f6e\u7684\u6570\u503c\uff0c\u7531\u514b\u62c9\u9ed8\u6cd5\u5219<\/p>\n<div class=\"katex math multi-line no-emojify\">\\{A^{-1}\u4e2d(i,j)\u5143\u7d20\\}=x_i=\\frac{\\det A_i(e_j)}{\\det A} \\tag{2}\n<\/div>\n<p>\u56de\u60f3\u8d77<span class=\"katex math inline\">A_{ji}<\/span>\u8868\u793aA\u7684\u5b50\u77e9\u9635\uff0c\u5b83\u7531A\u53bb\u6389\u7b2cj\u884c\u548c\u7b2ci\u884c\u5f97\u5230.<span class=\"katex math inline\">A_i(e_j)<\/span>\u6309\u7b2ci\u5217\u7684\u4f59\u56e0\u5b50\u5c55\u5f00\u5f0f\u4e3a<\/p>\n<div class=\"katex math multi-line no-emojify\">\\det A_i(e_j)=(-1)^{i+j}\\det A_{ji}=C_{ji} \\tag{3}\n<\/div>\n<p>\u8fd9\u91cc<span class=\"katex math inline\">C_{ji}<\/span>\u662fA\u7684\u4e00\u4e2a\u4f59\u56e0\u5b50\uff0c\u7531(2),<span class=\"katex math inline\">A^{-1}<\/span>\u7684\uff08i,j\uff09\u5143\u7d20\u7b49\u4e8e\u4f59\u56e0\u5b50<span class=\"katex math inline\">C_{ji}<\/span>\u9664\u4ee5<span class=\"katex math inline\">\\det A<\/span>.\uff08\u6ce8\u610f\uff1a<span class=\"katex math inline\">C_{ji}<\/span>\u7684\u4e0b\u8868\u662f(i,j)\u7684\u98a0\u5012\uff09\u4e8e\u662f<\/p>\n<div class=\"katex math multi-line no-emojify\">A^{-1}=\\frac{1}{\\det A}<br \/>\n\\begin{bmatrix}<br \/>\nC_{11}&amp;C_{21}&amp; \\dots &amp;C_{n1}\\\\<br \/>\nC_{12}&amp;C_{22}&amp;\\dots &amp; C_{n2}\\\\<br \/>\n\\vdots&amp;\\vdots&amp;&amp;\\vdots\\\\<br \/>\nC_{1n}&amp;C_{2n}&amp;\\dots &amp; C_{nn}\\\\<br \/>\n\\end{bmatrix} \\tag4\n<\/div>\n<p>\u516c\u5f0f\uff084\uff09\u53f3\u8fb9\u7684\u4f59\u56e0\u5b50\u7684\u77e9\u9635\u6210\u4e3aA\u7684\u4f34\u968f\u77e9\u9635\uff0c\u8bb0\u4e3aadj A\u3002(\u4f34\u968f\u8fd9\u4e2a\u672f\u8bed\u5728\u540e\u9762\u7684\u7ebf\u6027\u53d8\u6362\u8bfe\u7a0b\u4e2d\u8fd8\u6709\u53e6\u4e00\u5c42\u610f\u601d)<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u74068<\/strong> \u9006\u77e9\u9635\u516c\u5f0f<br \/>\n  \u8bbeA\u662f\u4e00\u4e2a\u53ef\u9006\u7684nxn\u77e9\u9635\uff0c\u5219<span class=\"katex math inline\">A^{-1}=\\frac{1}{\\det A}adj A<\/span>\n<\/p><\/blockquote>\n<p>#\u7528\u884c\u5217\u5f0f\u8868\u793a\u9762\u79ef\u6216\u4f53\u79ef<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u74069<\/strong> \u82e5A\u662f\u4e00\u4e2a2&#215;2\u77e9\u9635\uff0c\u5219\u7531A\u7684\u5217\u786e\u5b9a\u7684\u5e73\u884c\u56db\u8fb9\u5f62\u7684\u9762\u79ef\u4e3a<span class=\"katex math inline\">|\\det A|<\/span>,\u82e5A\u662f\u4e00\u4e2a3&#215;3\u77e9\u9635\uff0c\u5219\u7531A\u5f97\u5217\u786e\u5b9a\u7684\u5e73\u884c\u516d\u9762\u4f53\u7684\u4f53\u79ef\u4e3a<span class=\"katex math inline\">|\\det A|<\/span>\n<\/p><\/blockquote>\n<p>\u8bbe<span class=\"katex math inline\">a_1<\/span>\u548c<span class=\"katex math inline\">a_2<\/span>\u4e3a\u975e\u96f6\u5411\u91cf\uff0c\u5219\u5bf9\u4efb\u610f\u6570c,\u6709<span class=\"katex math inline\">a_1<\/span>\u548c<span class=\"katex math inline\">a_2<\/span>\u786e\u5b9a\u7684\u5e73\u884c\u56db\u8fb9\u5f62\u7684\u9762\u79ef\u7b49\u4e8e\u7531<span class=\"katex math inline\">a_1<\/span>\u548c<span class=\"katex math inline\">a_2+ca_1<\/span>\u786e\u5b9a\u7684\u5e73\u884c\u56db\u8fb9\u5f62\u7684\u9762\u79ef<\/p>\n<h1><span id=\"i\">\u7ebf\u6027\u53d8\u6362<\/span><\/h1>\n<p>\u884c\u5217\u5f0f\u53ef\u7528\u4e8e\u63cf\u8ff0\u5e73\u9762\u548c<span class=\"katex math inline\">R^3<\/span>\u4e2d\u7ebf\u6027\u53d8\u6362\u7684\u4e00\u4e2a\u91cd\u8981\u7684\u51e0\u4f55\u6027\u8d28\u3002\u82e5T\u662f\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\uff0cS\u662fT\u7684\u5b9a\u4e49\u57df\u5185\u7684\u4e00\u4e2a\u96c6\u5408\uff0c\u7528<span class=\"katex math inline\">T(S)<\/span>\u8868\u793aS\u4e2d\u70b9\u7684\u50cf\u96c6\u3002<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u740610<\/strong>\u3000\u8bbe<span class=\"katex math inline\">T:R^2 \\mapsto R^2<\/span>\u662f\u7531\u4e00\u4e2a2&#215;2\u77e9\u9635A\u786e\u5b9a\u7684\u7ebf\u6027\u53d8\u6362\uff0c\u82e5S\u662f<span class=\"katex math inline\">R^2<\/span>\u4e2d\u4e00\u4e2a\u5e73\u884c\u56db\u8fb9\u5f62\uff0c\u5219\n<\/p><\/blockquote>\n<div class=\"katex math multi-line no-emojify\">\\{T(s)\u7684\u9762\u79ef\\}=|\\det A|\\cdot\\{S\u7684\u9762\u79ef\\}   \\tag{5}\n<\/div>\n<p>\u82e5T\u662f\u4e00\u4e2a\u75313&#215;3\u77e9\u9635A\u786e\u5b9a\u7684\u7ebf\u6027\u53d8\u6362\uff0c\u800cS\u662f<span class=\"katex math inline\">R^3<\/span>\u4e2d\u7684\u4e00\u4e2a\u5e73\u884c\u516d\u9762\u4f53\uff0c\u5219<\/p>\n<div class=\"katex math multi-line no-emojify\">\\{T(S)\u7684\u4f53\u79ef\\}=|\\det A|\\cdot\\{S\u7684\u4f53\u79ef\\}   \\tag{6}\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Contents1 \u4e00\u4e2a\u6c42A^{-1}\u7684\u516c\u5f0f2 \u7ebf\u6027\u53d8\u6362 \u514b\u62c9\u9ed8\u6cd5\u5219\u5728\u5404\u79cd\u7406\u8bba\u8ba1\u7b97\u4e2d\u662f\u5fc5\u9700\u7684\uff0c\u4f8b\u5982\uff0c\u5b83\u88ab\u7528\u6765\u7814 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":369,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,22,26],"tags":[],"class_list":["post-367","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\/367","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=367"}],"version-history":[{"count":5,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/367\/revisions"}],"predecessor-version":[{"id":373,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/367\/revisions\/373"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/media\/369"}],"wp:attachment":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=367"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=367"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=367"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}