{"id":553,"date":"2021-05-16T21:01:58","date_gmt":"2021-05-16T13:01:58","guid":{"rendered":"http:\/\/www.wayln.com\/?p=553"},"modified":"2021-05-16T22:39:04","modified_gmt":"2021-05-16T14:39:04","slug":"5-4-%e7%89%b9%e5%be%81%e5%90%91%e9%87%8f%e4%b8%8e%e7%ba%bf%e6%80%a7%e5%8f%98%e6%8d%a2","status":"publish","type":"post","link":"https:\/\/www.wayln.com\/?p=553","title":{"rendered":"5.4 \u7279\u5f81\u5411\u91cf\u4e0e\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=\"#i\"><span class=\"toc_number toc_depth_1\">1<\/span> \u7ebf\u6027\u53d8\u6362\u7684\u77e9\u9635<\/a><\/li><li><a href=\"#VV\"><span class=\"toc_number toc_depth_1\">2<\/span> V\u5230V\u7684\u7ebf\u6027\u53d8\u6362<\/a><ul><li><a href=\"#Rn\"><span class=\"toc_number toc_depth_2\">2.1<\/span> $R^n$\u4e0a\u7684\u7ebf\u6027\u53d8\u6362<\/a><\/li><\/ul><\/li><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_1\">3<\/span> \u77e9\u9635\u8868\u793a\u7684\u76f8\u4f3c\u6027<\/a><\/li><\/ul><\/div>\n<p>\u5728\u672c\u8282\uff0c\u6211\u4eec\u5c06\u77e9\u9635\u5206\u89e3<span class=\"katex math inline\">A=PDP^{-1}<\/span>\u7406\u89e3\u4e3a\u7ebf\u6027\u53d8\u6362\u3002\u6211\u4eec\u8fd8\u5c06\u770b\u5230\u53d8\u6362<span class=\"katex math inline\">x \\mapsto Ax<\/span>\u5b9e\u8d28\u4e0a\u662f\u7b80\u5355\u7684\u6620\u5c04<span class=\"katex math inline\">u \\mapsto Du<\/span>\u3002\u5373\u4f7fD\u4e0d\u662f\u5bf9\u89d2\u77e9\u9635\u65f6\uff0c\u5bf9\u77e9\u9635A\u548cD\u4ecd\u6709\u76f8\u4f3c\u7684\u89e3\u91ca\u3002<br \/>\n\u57281.9\u8282\u66fe\u8bb2\u5230\uff0c\u4efb\u610f\u4e00\u4e2a\u4ece<span class=\"katex math inline\">R^n\u5230R^m<\/span>\u7684\u7ebf\u6027\u53d8\u6362T\u53ef\u901a\u8fc7\u5de6\u4e58\u77e9\u9635A\u6765\u5b9e\u73b0\uff0c\u77e9\u9635A\u6210\u4e3aT\u7684\u6807\u51c6\u77e9\u9635\u3002\u73b0\u5728\uff0c\u6211\u4eec\u5bf9\u8fde\u4e2a\u6709\u9650\u7ef4\u5411\u91cf\u7a7a\u95f4\u4e4b\u95f4\u7684\u7ebf\u6027\u53d8\u6362\u4e5f\u505a\u540c\u6837\u7684\u63cf\u8ff0\u3002<\/p>\n<h1><span id=\"i\">\u7ebf\u6027\u53d8\u6362\u7684\u77e9\u9635<\/span><\/h1>\n<p>\u8bbeV\u662fn\u7ef4\u5411\u91cf\u7a7a\u95f4\uff0cW\u662fm\u7ef4\u5411\u91cf\u7a7a\u95f4\uff0cT\u662fV\u5230W\u7684\u7ebf\u6027\u53d8\u6362\uff0c\u4e3a\u4e86\u628aT\u4e0e\u77e9\u9635\u76f8\u8054\u7cfb\uff0c\u6211\u4eec\u6307\u5b9a<span class=\"katex math inline\">\\beta\u548cC<\/span>\u5206\u522b\u662fV\u548cW\u7684\u57fa\u3002<br \/>\n\u82e5<span class=\"katex math inline\">x \\in V<\/span>,\u5750\u6807\u5411\u91cf<span class=\"katex math inline\">\\begin{bmatrix} x \\end{bmatrix}&#95;\\beta  \\in R^n<\/span>\uff0cx\u7684\u50cf<span class=\"katex math inline\">T(x)<\/span>\u7684\u5750\u6807\u5411\u91cf<span class=\"katex math inline\">\\begin{bmatrix} T(x) \\end{bmatrix}&#95;C \\in R^m<\/span>\uff0c\u89c1\u4e0b\u56fe\uff0c<br \/>\n\u5bb9\u6613\u627e\u51fa<span class=\"katex math inline\">\\begin{bmatrix}x\\end{bmatrix}_\\beta<\/span>\u548c<span class=\"katex math inline\">\\begin{bmatrix}T(x)\\end{bmatrix}_C<\/span>\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u8bbeV\u7684\u57fa<span class=\"katex math inline\">\\beta<\/span>\u662f<span class=\"katex math inline\">&#92;{b_1,\\dots,b_n&#92;}<\/span>\u3002\u82e5<span class=\"katex math inline\">x=r_1b_1+\\dots+r_nb_n<\/span>,\u5219<\/p>\n<div class=\"katex math multi-line no-emojify\">\\begin{bmatrix}x\\end{bmatrix}_\\beta=\\begin{bmatrix}r_1\\\\ \\vdots\\\\ r_n \\end{bmatrix}\n<\/div>\n<p>\u56e0\u4e3aT\u662f\u7ebf\u6027\u7684\uff0c\u6545<\/p>\n<div class=\"katex math multi-line no-emojify\">T(x)=T(r_1b_1+\\dots+r_nb_n)=r_1T(b_1)+\\dots+r_nT(b_n) \\tag{1}\n<\/div>\n<p>\u56e0\u4e3a\u4eceW\u5230<span class=\"katex math inline\">R^m<\/span>\u7684\u5750\u6807\u6620\u5c04\u662f\u7ebf\u6027\u7684\uff084.4\u8282\u5b9a\u74068\uff09\uff0c\u7b49\u5f0f\uff081\uff09\u53ef\u63a8\u51fa<\/p>\n<div class=\"katex math multi-line no-emojify\">\\left[T(x)\\right]_C=r_1\\left[T(b_1)\\right]_C+\\dots+r_n\\left[T(b_n)\\right] \\tag{2}\n<\/div>\n<p><a class=\"wp-editor-md-post-content-link\" href=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/05\/wp_editor_md_d36a31d291e698e7bc227a62a259d31a.jpg\"><img decoding=\"async\" src=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/05\/wp_editor_md_d36a31d291e698e7bc227a62a259d31a.jpg\" alt=\"\" \/><\/a><br \/>\n\u56e0\u4e3a\u8fd9\u4e9bC-\u5750\u6807\u5411\u91cf\u90fd\u5c5e\u4e8e<span class=\"katex math inline\">R^m<\/span>,\u5411\u91cf\u7b49\u5f0f\uff082\uff09\u53ef\u4ee5\u5199\u4e3a\u77e9\u9635\u7b49\u5f0f<\/p>\n<div class=\"katex math multi-line no-emojify\">\\left[T(x)\\right]_C=M\\left[x\\right]_\\beta \\tag{3}\n<\/div>\n<p>\u5176\u4e2d<\/p>\n<div class=\"katex math multi-line no-emojify\">M=\\begin{bmatrix}\\left[T(b_1)\\right]_C&amp;\\left[T(b)_2\\right]_C&amp;\\dots &amp;\\left[T(b_n)\\right]_C\\end{bmatrix} \\tag{4}\n<\/div>\n<p>\u77e9\u9635M\u662fT\u7684\u77e9\u9635\u8868\u793a\uff0c\u79f0\u4e3a<strong>T\u76f8\u5bf9\u4e8e\u57fa<span class=\"katex math inline\">\\beta<\/span>\u548cC\u7684\u77e9\u9635<\/strong>\uff0c\u89c1\u4e0b\u56fe<br \/>\n<a class=\"wp-editor-md-post-content-link\" href=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/05\/wp_editor_md_97c3472da7af85a5442159e09cb51f22.jpg\"><img decoding=\"async\" src=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/05\/wp_editor_md_97c3472da7af85a5442159e09cb51f22.jpg\" alt=\"\" \/><\/a><br \/>\n\u7b49\u5f0f\uff083\uff09\u8868\u660e\uff0c\u5c31\u5750\u6807\u5411\u91cf\u800c\u8a00\uff0cT\u5bf9x\u7684\u4f5c\u7528\u76f8\u5f53\u4e8e\u7528\u77e9\u9635M\u5de6\u4e58x<\/p>\n<h1><span id=\"VV\">V\u5230V\u7684\u7ebf\u6027\u53d8\u6362<\/span><\/h1>\n<p>\u5f53W=V\uff0cC=<span class=\"katex math inline\">\\beta<\/span>\u65f6\uff0c\uff084\uff09\u4e2d\u7684M\u79f0\u4e3aT\u76f8\u5bf9\u4e8e<span class=\"katex math inline\">\\beta<\/span>\u7684\u77e9\u9635\uff0c\u6216\u7b80\u79f0\u4e3aT\u7684<span class=\"katex math inline\">\\beta-<\/span>\u77e9\u9635\uff0c\u8bb0\u4e3a<span class=\"katex math inline\">\\left[T\\right]_\\beta<\/span>,\u89c1\u4e0b\u56fe\uff1a<br \/>\n<a class=\"wp-editor-md-post-content-link\" href=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/05\/wp_editor_md_4af498e6efa1f69b15204e0f6ce4a75e.jpg\"><img decoding=\"async\" src=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/05\/wp_editor_md_4af498e6efa1f69b15204e0f6ce4a75e.jpg\" alt=\"\" \/><\/a><br \/>\n<span class=\"katex math inline\">V \\rightarrow V<\/span>\u7684\u7ebf\u6027\u53d8\u6362T\u7684<span class=\"katex math inline\">\\beta-<\/span>\u77e9\u9635\u5bf9\u6240\u6709V\u4e2d\u7684x,\u6709<\/p>\n<div class=\"katex math multi-line no-emojify\">\\left[T(x)\\right]_\\beta=\\left[T\\right]_\\beta\\left[x\\right]_\\beta \\tag{5}\n<\/div>\n<h2><span id=\"Rn\">$R^n$\u4e0a\u7684\u7ebf\u6027\u53d8\u6362<\/span><\/h2>\n<p>\u5728\u6d89\u53ca<span class=\"katex math inline\">R^n<\/span>\u7684\u5e94\u7528\u95ee\u9898\u4e2d\uff0c\u7ebf\u6027\u53d8\u6362\u9996\u5148\u8868\u73b0\u4e3a\u4e00\u4e2a\u77e9\u9635\u53d8\u6362<span class=\"katex math inline\">x \\mapsto Ax<\/span>\u3002\u5047\u8bbeA\u662f\u53ef\u5bf9\u89d2\u5316\u7684\uff0c\u90a3\u4e48\u5b58\u5728\u7531A\u7684\u7279\u5f81\u5411\u91cf\u7ec4\u6210\u7684<span class=\"katex math inline\">R^n<\/span>\u7684\u57fa\u3002\u6b64\u65f6\uff0c\u4e0b\u9762\u7684\u5b9a\u74068\u8868\u660eT\u7684<span class=\"katex math inline\">\\beta-<\/span>\u77e9\u9635\u662f\u5bf9\u89d2\u77e9\u9635\uff0c\u8fd9\u6837\uff0c\u628aA\u5bf9\u89d2\u5316\u76f8\u5f53\u4e8e\u627e\u5230\u53d8\u6362<span class=\"katex math inline\">x \\mapsto Ax<\/span>\u7684\u5bf9\u89d2\u77e9\u9635\u8868\u793a\u3002<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u74068<\/strong>\uff08\u5bf9\u89d2\u77e9\u9635\u8868\u793a\uff09<br \/>\n  \u8bbe<span class=\"katex math inline\">A=PDP^{-1}<\/span>\uff0c\u5176\u4e2dD\u4e3anxn\u5bf9\u89d2\u77e9\u9635\uff0c\u82e5<span class=\"katex math inline\">R^n<\/span>\u7684\u57fa<span class=\"katex math inline\">\\beta<\/span>\u7531P\u7684\u5217\u5411\u91cf\u7ec4\u6210\uff0c\u90a3\u4e48D\u662f\u53d8\u6362<span class=\"katex math inline\">x \\mapsto Ax<\/span>\u7684<span class=\"katex math inline\">\\beta-<\/span>\u77e9\u9635\n<\/p><\/blockquote>\n<h1><span id=\"i-2\">\u77e9\u9635\u8868\u793a\u7684\u76f8\u4f3c\u6027<\/span><\/h1>\n<p>\u5b9a\u74068\u7684\u8bc1\u660e\u6ca1\u6709\u7528\u5230D\u662f\u5bf9\u89d2\u77e9\u9635\u8fd9\u4e00\u4e8b\u5b9e\u3002\u56e0\u6b64\uff0c\u82e5A\u76f8\u4f3c\u4e8eC\uff0c\u5373\u7531<span class=\"katex math inline\">A=PCP^{-1}<\/span>,\u5982\u679c<span class=\"katex math inline\">\\beta<\/span>\u7531P\u7684\u5217\u5411\u91cf\u7ec4\u6210\uff0c\u5219C\u662f\u53d8\u6362<span class=\"katex math inline\">x \\mapsto Ax<\/span>\u7684<span class=\"katex math inline\">\\beta-<\/span>\u77e9\u9635\u3002\u5206\u89e3<span class=\"katex math inline\">A=PCP^{-1}<\/span>\u5982\u4e0b\u56fe\u6240\u793a\uff1a<br \/>\n<a class=\"wp-editor-md-post-content-link\" href=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/05\/wp_editor_md_62ab7499cdf209a2cd6ff2d12971035e.jpg\"><img decoding=\"async\" src=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/05\/wp_editor_md_62ab7499cdf209a2cd6ff2d12971035e.jpg\" alt=\"\" \/><\/a><br \/>\n\u76f8\u53cd\uff0c\u82e5<span class=\"katex math inline\">R^n \\rightarrow R^n<\/span>\u7684\u53d8\u6362<span class=\"katex math inline\">T:T(x)=Ax<\/span>,\u800c<span class=\"katex math inline\">\\beta\u662fR^n<\/span>\u7684\u4efb\u610f\u4e00\u4e2a\u57fa\uff0c\u90a3\u4e48T\u7684<span class=\"katex math inline\">\\beta-<\/span>\u77e9\u9635\u76f8\u4f3c\u4e8eA<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Contents1 \u7ebf\u6027\u53d8\u6362\u7684\u77e9\u96352 V\u5230V\u7684\u7ebf\u6027\u53d8\u63622.1 $R^n$\u4e0a\u7684\u7ebf\u6027\u53d8\u63623 \u77e9\u9635\u8868\u793a\u7684\u76f8\u4f3c\u6027 \u5728\u672c [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":332,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,22,26],"tags":[],"class_list":["post-553","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\/553","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=553"}],"version-history":[{"count":12,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/553\/revisions"}],"predecessor-version":[{"id":565,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/553\/revisions\/565"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/media\/332"}],"wp:attachment":[{"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=553"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=553"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=553"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}