{"id":237,"date":"2016-08-22T22:11:39","date_gmt":"2016-08-22T22:11:39","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=237"},"modified":"2016-12-21T05:29:58","modified_gmt":"2016-12-21T05:29:58","slug":"guan-yu-liang-ge-ji-di-de-hun-he-ji-yu-xing-lie-shi-de-guan-xi","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=237","title":{"rendered":"\u5173\u4e8e\u4e24\u4e2a\u57fa\u5e95\u7684\u6df7\u5408\u79ef\u4e0e\u884c\u5217\u5f0f\u7684\u5173\u7cfb"},"content":{"rendered":"

\u6211\u4eec\u6709\u5982\u4e0b\u57fa\u672c\u4e8b\u5b9e:
\n

\u5b9a\u7406 1<\/span>.<\/span> \u5047\u8bbe$\\set{v_1,v_2,\\ldots,v_n}$\u548c$w_1,w_2,\\ldots,w_n$\u662f$\\RR^n$\u4e2d\u4e24\u4e2a\u57fa, \u5b9a\u4e49$(n-1)\\times(n-1)$\u77e9\u9635
\n$$
\nA=\\begin{pmatrix}\\inner{v_i,w_j}\\end{pmatrix}
\n=\\begin{pmatrix}
\n v_1\\cdot w_1&v_1\\cdot w_2&\\cdots&v_1\\cdot w_{n-1}\\\\
\n v_2\\cdot w_1&v_2\\cdot w_2&\\cdots&v_2\\cdot w_{n-1}\\\\
\n \\vdots &\\vdots &&\\vdots\\\\
\n v_{n-1}\\cdot w_1&v_{n-1}\\cdot w_2&\\cdots&v_{n-1}\\cdots w_{n-1}
\n\\end{pmatrix},
\n$$
\n\u5219
\n\\begin{equation}\\label{eq:res}
\n\\inner{v_1\\wedge v_2\\wedge\\cdots\\wedge v_{n-1},w_1\\wedge w_2\\wedge\\cdots\\wedge w_{n-1}}=\\det A.
\n\\end{equation}
\n<\/div>
\n<\/p>\n
\u8bc1\u660e<\/span>.<\/span> \u9996\u5148, \u6ce8\u610f\u5230\\eqref{eq:res}\u4e24\u8fb9\u5173\u4e8e\u4e09\u79cd\u521d\u7b49\u53d8\u6362(\u6570\u4e58\u4e00\u884c,\u4ea4\u6362\u4e24\u884c,\u67d0\u884c\u7684\u500d\u6570\u52a0\u5230\u53e6\u4e00\u884c)\u662f\u6210\u7acb\u7684. \u6545\u6211\u4eec\u53ea\u9700\u8bc1\u660e$\\set{v_1,v_2,\\ldots,v_n}$\u662f\u6807\u51c6\u6b63\u4ea4\u57fa$\\set{e_1,e_2,\\ldots,e_n}$\u7684\u60c5\u5f62.<\/p>\n

\u8bb0\u5217\u5411\u91cf$w_1,w_2,\\ldots,w_n$\u6784\u6210\u7684\u77e9\u9635\u4e3a$W=\\begin{pmatrix}w_1&w_2&\\cdots&w_n\\end{pmatrix}$, \u5e76\u5c06\u5176\u5206\u5757\u4e3a$W=\\begin{pmatrix}W_1\\\\W_0\\end{pmatrix}$. \u5219
\n$$
\nA=\\begin{pmatrix}\\inner{e_i,w_j}\\end{pmatrix}=W_1.
\n$$<\/p>\n

\u53e6\u4e00\u65b9\u9762, \u6309\u7167\u5916\u79ef\u7684\u5b9a\u4e49(\u5982\u4e0b\u77e9\u9635\u6309\u6700\u540e\u4e00\u884c\u5c55\u5f00)
\n$$
\nw_1\\wedge w_2\\wedge\\cdots\\wedge w_{n-1}=\\det\\begin{pmatrix}
\n&w_1^T\\\\
\n&w_2^T\\\\
\n&\\vdots\\\\
\n&w_{n-1}^T\\\\
\ne_1&\\cdots &e_{n-1}&e_n
\n\\end{pmatrix}\\\\
\n=\\sum_{i=1}^{n-1}\\lambda_ie_i+\\det W_1^Te_n.
\n$$
\n\u53ef\u89c1
\n$$
\n\\inner{v_1\\wedge v_2\\wedge\\cdots\\wedge v_{n-1},w_1\\wedge w_2\\wedge\\cdots\\wedge w_{n-1}}=\\inner{e_n,w_1\\wedge w_2\\wedge\\cdots\\wedge w_{n-1}}=\\det W_1^T=\\det W_1=\\det A.
\n$$
\n<\/div>\n","protected":false},"excerpt":{"rendered":"

\u6211\u4eec\u6709\u5982\u4e0b\u57fa\u672c\u4e8b\u5b9e: \u5b9a\u7406 1. \u5047\u8bbe$\\set{v_1,v_2,\\ldots,v_n}$\u548c$w_1,w_2,… \u7ee7\u7eed\u9605\u8bfb\u5173\u4e8e\u4e24\u4e2a\u57fa\u5e95\u7684\u6df7\u5408\u79ef\u4e0e\u884c\u5217\u5f0f\u7684\u5173\u7cfb<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[25,29,41,46],"class_list":["post-237","post","type-post","status-publish","format-standard","hentry","category-math","tag-nei-ji","tag-wai-ji","tag-hun-he-ji","tag-xing-lie-shi","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/237","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=237"}],"version-history":[{"count":1,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/237\/revisions"}],"predecessor-version":[{"id":272,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/237\/revisions\/272"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=237"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=237"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=237"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}