{"id":260,"date":"2021-04-10T21:05:50","date_gmt":"2021-04-10T13:05:50","guid":{"rendered":"http:\/\/www.wayln.com\/?p=260"},"modified":"2021-04-15T10:49:53","modified_gmt":"2021-04-15T02:49:53","slug":"2-2-%e7%9f%a9%e9%98%b5%e7%9a%84%e9%80%86","status":"publish","type":"post","link":"http:\/\/www.wayln.com\/?p=260","title":{"rendered":"2.2 \u77e9\u9635\u7684\u9006"},"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> \u521d\u7b49\u77e9\u9635<\/a><\/li><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_1\">2<\/span> \u9006\u77e9\u9635\u7684\u53e6\u4e00\u4e2a\u89c2\u70b9<\/a><\/li><\/ul><\/div>\n<p>\u5b9e\u65705\u7684\u4e58\u6cd5\u9006\u662f<span class=\"katex math inline\">1\/5<\/span>\u6216<span class=\"katex math inline\">5^{-1}<\/span>,\u5b83\u6ee1\u8db3\u65b9\u7a0b<\/p>\n<div class=\"katex math multi-line no-emojify\">5^{-1}\\cdot5=1\u3000\u548c\u30005 \\cdot 5^{-1}=1\n<\/div>\n<p>\u3000\u77e9\u9635\u5bf9\u9006\u7684\u4e00\u822c\u5316\u4e5f\u8981\u6c42\u4e24\u4e2a\u65b9\u7a0b\u540c\u65f6\u6210\u7acb\uff0c\u5e76\u907f\u514d\u4f7f\u7528\u659c\u7ebf\u8bb0\u53f7\u8868\u793a\u9664\u6cd5\uff0c\u56e0\u4e3a\u77e9\u9635\u4e58\u6cd5\u4e0d\u662f\u53ef\u4ea4\u6362\u7684\uff0c\u8fdb\u4e00\u6b65\uff0c\u5b8c\u5168\u7684\u4e00\u822c\u5316\u662f\u53ef\u80fd\u7684\uff0c\u5f53\u4e14\u4ec5\u5f53\u6709\u5173\u77e9\u9635\u662f\u65b9\u9635\u3002<br \/>\n\u4e00\u4e2anxn\u77e9\u9635<span class=\"katex math inline\">A<\/span>\u662f\u53ef\u9006\u7684\uff0c\u82e5\u36ee\u4e00\u4e2anxn\u77e9\u9635<span class=\"katex math inline\">C<\/span>\u4f7f\uff1a<\/p>\n<div class=\"katex math multi-line no-emojify\">AC=I \u3000\u4e14 \u3000CA=I\n<\/div>\n<p>\u8fd9\u91cc<span class=\"katex math inline\">I=I_n<\/span>\u662fnxn\u5355\u4f4d\u77e9\u9635\uff0c\u8fd9\u65f6\u79f0<span class=\"katex math inline\">C<\/span>\u662f<span class=\"katex math inline\">A<\/span>\u7684\u9006\u9635\u3002\u5b9e\u9645\u4e0a\uff0c<span class=\"katex math inline\">C<\/span>\u7531<span class=\"katex math inline\">A<\/span>\u552f\u4e00\u786e\u5b9a\uff0c\u56e0\u4e3a\u82e5<span class=\"katex math inline\">B<\/span>\u662f\u53e6\u4e00\u4e2a<span class=\"katex math inline\">A<\/span>\u7684\u9006\u9635\uff0c\u90a3\u4e48\u5c06\u6709<span class=\"katex math inline\">B=BI=B(AC)=(BA)C=IC=C<\/span>,\u4e8e\u662f\uff0c\u82e5<span class=\"katex math inline\">A<\/span>\u53ef\u9006\uff0c\u5b83\u7684\u9006\u662f\u552f\u4e00\u7684\uff0c\u6211\u4eec\u5c06\u5b83\u8bb0\u4e3a<span class=\"katex math inline\">A^{-1}<\/span>\uff0c\u4e8e\u662f<\/p>\n<div class=\"katex math multi-line no-emojify\">AA^{-1}=I\u3000\u4e14\u3000A^{-1}A=I\n<\/div>\n<p>\u4e0d\u53ef\u9006\u7684\u77e9\u9635\u6709\u65f6\u79f0\u4e3a<strong>\u5947\u5f02\u77e9\u9635<\/strong>,\u800c\u53ef\u9006\u77e9\u9635\u4e5f\u79f0\u4e3a<strong>\u975e\u5947\u5f02\u77e9\u9635<\/strong><\/p>\n<blockquote><p>\n  <strong>\u5b9a\u74064<\/strong>\u3000\u8bbe<span class=\"katex math inline\">A=\\begin{bmatrix}a&amp;b&#92;\\c&amp;d\\end{bmatrix}<\/span>,\u82e5<span class=\"katex math inline\">ad-bc \\neq 0<\/span>\uff0c\u5219A\u53ef\u9006\uff0c\u4e14\n<\/p><\/blockquote>\n<div class=\"katex math multi-line no-emojify\">A^{-1}=\\frac{1}{ad-bc}\\left[\\begin{array}{rr}d&amp;-b\\\\-c&amp;a\\end{array}\\right]\n<\/div>\n<p>\u82e5<span class=\"katex math inline\">ad-bc=0<\/span>,\u5219A\u4e0d\u53ef\u9006<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u74065<\/strong>\u3000\u82e5<span class=\"katex math inline\">A<\/span>\u662f\u53ef\u9006nxn\u77e9\u9635\uff0c\u5219\u5bf9\u6bcf\u4e00<span class=\"katex math inline\">R^n<\/span>\u4e2d\u7684<span class=\"katex math inline\">b<\/span>,\u65b9\u7a0b<span class=\"katex math inline\">Ax=b<\/span>\u6709\u552f\u4e00\u89e3<span class=\"katex math inline\">x=A^{-1}b<\/span>\u3002\n<\/p><\/blockquote>\n<p>\u5b9a\u74065\u7684\u516c\u5f0f\u5f88\u5c11\u7528\u6765\u89e3\u65b9\u7a0b<span class=\"katex math inline\">Ax=b<\/span>,\u56e0\u4e3a<span class=\"katex math inline\">&#92;begin{bmatrix}A&amp;b&#92;end{bmatrix}<\/span>\u7684\u53d8\u6362\u901a\u5e38\u66f4\u5feb\u3002<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u74066<\/strong><br \/>\n  a.\u3000\u82e5<span class=\"katex math inline\">A<\/span>\u662f\u53ef\u9006\u77e9\u9635\uff0c\u5219<span class=\"katex math inline\">A^{-1}\u4e5f\u53ef\u9006\u800c\u4e14(A^{-1})^{-1}=A<\/span>\u3002<br \/>\n  b.\u3000\u82e5<span class=\"katex math inline\">A<\/span>\u548c<span class=\"katex math inline\">B<\/span>\u90fd\u662fnxn\u53ef\u9006\u77e9\u9635\uff0cAB\u4e5f\u53ef\u9006\uff0c\u4e14\u5176\u9006\u662fA\u548cB\u7684\u9006\u77e9\u9635\u6309\u76f8\u53cd\u987a\u5e8f\u7684\u4e58\u79ef\uff0c\u5373<\/p>\n<div class=\"katex math multi-line no-emojify\">(AB)^{-1}=B^{-1}A^{-1}\n  <\/div>\n<p>  c.\u3000\u82e5A\u53ef\u9006\uff0c\u5219<span class=\"katex math inline\">A^T<\/span>\u4e5f\u53ef\u9006\uff0c\u4e14\u5176\u9006\u662f<span class=\"katex math inline\">A^{-1}<\/span>\u7684\u8f6c\u7f6e\uff0c\u5373<span class=\"katex math inline\">\uff08A^T\uff09^{-1}=\uff08A^{-1}\uff09^T<\/span><\/p>\n<p>  \u82e5\u5e72\u4e2anxn\u53ef\u9006\u77e9\u9635\u7684\u79ef\u4e5f\u662f\u53ef\u9006\u7684\uff0c\u5176\u9006\u7b49\u4e8e\u8fd9\u4e9b\u77e9\u9635\u6309\u76f8\u53cd\u987a\u5e8f\u7684\u4e58\u79ef\n<\/p><\/blockquote>\n<h2><span id=\"i\">\u521d\u7b49\u77e9\u9635<\/span><\/h2>\n<p>\u628a\u5355\u4f4d\u77e9\u9635\u8fdb\u884c\u4e00\u6b21\u884c\u53d8\u6362\uff0c\u5c31\u5f97\u5230<strong>\u521d\u7b49\u77e9\u9635<\/strong><\/p>\n<blockquote><p>\n  \u3000\u82e5\u5bf9mxn\u77e9\u9635<span class=\"katex math inline\">A<\/span>\u8fdb\u884c\u67d0\u79cd\u521d\u7b49\u884c\u53d8\u6362\uff0c\u6240\u5f97\u77e9\u9635\u53ef\u5199\u6210<span class=\"katex math inline\">EA<\/span>\uff0c\u5176\u4e2d<span class=\"katex math inline\">E<\/span>\u662fmxm\u77e9\u9635\uff0c\u662f\u7531<span class=\"katex math inline\">I_m<\/span>\u8fdb\u884c\u540c\u4e00\u884c\u53d8\u6362\u6240\u5f97\u3002\n<\/p><\/blockquote>\n<p>\u82e5E\u662f\u7531I\u8fdb\u884c\u53d8\u6362\u6240\u5f97\uff0c\u5219\u6709\u540c\u4e00\u7c7b\u578b\u7684\u53e6\u4e00\u884c\u53d8\u6362\u628a<span class=\"katex math inline\">E<\/span>\u53d8\u56de<span class=\"katex math inline\">I<\/span>\uff0c\u56e0\u6b64\uff0c\u6709\u521d\u7b49\u77e9\u9635<span class=\"katex math inline\">F<\/span>\u4f7f<span class=\"katex math inline\">FE=I<\/span>.\u56e0<span class=\"katex math inline\">E<\/span>\u548c<span class=\"katex math inline\">F<\/span>\u5bf9\u5e94\u4e92\u9006\u7684\u53d8\u6362\uff0c\u6240\u4ee5\u4e5f\u6709<span class=\"katex math inline\">EF=I<\/span>.<\/p>\n<blockquote><p>\n  \u3000\u6bcf\u4e2a\u521d\u7b49\u77e9\u9635<span class=\"katex math inline\">E<\/span>\u662f\u53ef\u9006\u7684\uff0c<span class=\"katex math inline\">E<\/span>\u7684\u9006\u662f\u4e00\u4e2a\u540c\u7c7b\u578b\u7684\u521d\u7b49\u77e9\u9635\uff0c\u5b83\u628a<span class=\"katex math inline\">E<\/span>\u53d8\u56de<span class=\"katex math inline\">I<\/span>\u3002<\/p>\n<p>  <strong>\u5b9a\u74067<\/strong>\u3000nxn\u77e9\u9635<span class=\"katex math inline\">A<\/span>\u662f\u53ef\u9006\u7684\uff0c\u5f53\u4e14\u4ec5\u5f53<span class=\"katex math inline\">A<\/span>\u884c\u7b49\u4ef7\u4e8e<span class=\"katex math inline\">I_n<\/span>\uff0c\u8fd9\u662f\uff0c\u628a<span class=\"katex math inline\">A<\/span>\u53d8\u4e3a<span class=\"katex math inline\">I_n<\/span>\u7684\u4e00\u7cfb\u5217\u521d\u7b49\u884c\u53d8\u6362\u540c\u65f6\u628a<span class=\"katex math inline\">I_n<\/span>\u53d8\u6210<span class=\"katex math inline\">A^{-1}<\/span>\u3002\n<\/p><\/blockquote>\n<p>\u6c42<span class=\"katex math inline\">A^&#123;-1&#125;<\/span>\u7684\u7b97\u6cd5<br \/>\n \u628a\u589e\u5e7f\u77e9\u9635<span class=\"katex math inline\">\\begin{bmatrix}A&amp;I\\end{bmatrix}<\/span>\u8fdb\u884c\u884c\u5316\u7b80\uff0c\u82e5<span class=\"katex math inline\">A<\/span>\u884c\u7b49\u4ef7\u4e8e<span class=\"katex math inline\">I<\/span>\uff0c\u5219<span class=\"katex math inline\">\\begin{bmatrix}I&amp;A^{-1}\\end{bmatrix}<\/span>,\u5426\u5219\uff0cA\u6ca1\u6709\u9006\u3002<\/p>\n<h2><span id=\"i-2\">\u9006\u77e9\u9635\u7684\u53e6\u4e00\u4e2a\u89c2\u70b9<\/span><\/h2>\n<p>\u7528<span class=\"katex math inline\">e_1,\\dots,e_n<\/span>\u8868\u793a<span class=\"katex math inline\">I_n<\/span>\u7684\u5404\u5217\uff0c\u5219\u628a<span class=\"katex math inline\">\\begin{bmatrix} A&amp;I&#92;end{bmatrix}<\/span>\u884c\u53d8\u6362\u6210<span class=\"katex math inline\">\\begin{bmatrix}I&amp;A^{-1}\\end{bmatrix}<\/span>\u7684\u8fc7\u7a0b\u53ef\u770b\u505a\u89e3n\u4e2a\u65b9\u7a0b\u7ec4<\/p>\n<div class=\"katex math multi-line no-emojify\">Ax=e_1,\u3000Ax=e_2,\u3000\\dots,\u3000Ax=e_n \\tag{2}\n<\/div>\n<p>\u5176\u4e2d\u8fd9\u4e9b\u65b9\u7a0b\u7ec4\u7684\u201c\u589e\u5e7f\u5217\u201d\u90fd\u653e\u5728A\u7684\u53f3\u8fb9\uff0c\u6784\u6210\u77e9\u9635<\/p>\n<div class=\"katex math multi-line no-emojify\">\\begin{bmatrix}A&amp;e_1&amp;e_2&amp;\\dots&amp;e_n\\end{bmatrix}=\\begin{bmatrix}A&amp;I\\end{bmatrix}\n<\/div>\n<p>\u65b9\u7a0b<span class=\"katex math inline\">AA^{-1}=I<\/span>\u53ca\u77e9\u9635\u4e58\u6cd5\u5b9a\u4e49\u8bf4\u660e<span class=\"katex math inline\">A^{-1}<\/span>\u7684\u5217\u6b63\u597d\u662f\u65b9\u7a0b(2)\u7684\u89e3\u3002\u8fd9\u4e00\u70b9\u5f88\u6709\u7528\uff0c\u56e0\u4e3a\u5728\u67d0\u4e9b\u5e94\u7528\u95ee\u9898\u4e2d\uff0c\u53ea\u9700\u8981<span class=\"katex math inline\">A^{-1}<\/span>\u7684\u4e00\u5217\u6216\u4e24\u5217\uff0c\u8fd9\u65f6\u53ea\u9700\u8981\u89e3\uff082\uff09\u4e2d\u7684\u76f8\u5e94\u65b9\u7a0b<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Contents1 \u521d\u7b49\u77e9\u96352 \u9006\u77e9\u9635\u7684\u53e6\u4e00\u4e2a\u89c2\u70b9 \u5b9e\u65705\u7684\u4e58\u6cd5\u9006\u662f1\/5\u62165^{-1},\u5b83\u6ee1\u8db3\u65b9\u7a0b 5^{- [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":347,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,22,26],"tags":[],"class_list":["post-260","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-21","category-22","category-26"],"_links":{"self":[{"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/260","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=260"}],"version-history":[{"count":7,"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/260\/revisions"}],"predecessor-version":[{"id":267,"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/260\/revisions\/267"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/media\/347"}],"wp:attachment":[{"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=260"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=260"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=260"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}