{"id":803,"date":"2021-07-16T01:21:35","date_gmt":"2021-07-16T01:21:35","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=803"},"modified":"2021-07-16T02:45:35","modified_gmt":"2021-07-16T02:45:35","slug":"liouvilledingliyuteshuxianxingqundelidaishu","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=803","title":{"rendered":"Liouville\u5b9a\u7406\u4e0e\u7279\u6b8a\u7ebf\u6027\u7fa4\u7684\u674e\u4ee3\u6570"},"content":{"rendered":"

\u5b9a\u7406 1<\/span>.<\/span> \u5047\u8bbe$A(t)$\u662f$n\\times n$\u7684\u5355\u53c2\u6570$n$\u9636\u65b9\u9635, $t\\in(-\\epsilon,\\epsilon)$. \u82e5$A(t)$\u662f\u7531$X(t)$\u751f\u6210\u7684, \u5373\u6ee1\u8db3
\n \\[
\n A'(t)=X(t)A(t),\\quad\\forall t\\in(-\\epsilon,\\epsilon),
\n \\]
\n \u5219
\n \\[
\n \\left( \\det A(t) \\right)’=\\mathrm{tr}\\left( X(t) \\right)\\det A(t).
\n \\]
\n<\/div>
\n
\n
\u8bc1\u660e<\/span>.<\/span> \u56de\u5fc6, \u5bf9\u77e9\u9635\u884c\u5217\u5f0f\u6c42\u5bfc\u7684Leibniz\u6cd5\u5219:
\n \\[
\n (\\det A(t))’=\\sum_{i=1}^n\\det
\n \\begin{pmatrix}
\n a_{11}&a_{12}&\\cdots&a_{1n}\\\\
\n \\vdots&\\vdots&&\\vdots\\\\
\n a_{i1}’&a_{i2}’&\\cdots&a_{in}’\\\\
\n \\vdots&\\vdots&&\\vdots\\\\
\n a_{n1}&a_{n2}&\\cdots&a_{nn}
\n \\end{pmatrix}.
\n \\]
\n \u56e0\u6b64, \u5982\u679c$A'(t)=X(t)A(t)$, \u90a3\u4e48$a_{ij}’=\\sum_{k=1}^nx_{ik}a_{kj}$, \u4ece\u800c
\n \\[
\n \\begin{aligned}
\n (\\det A(t))’&=\\sum_{i=1}^n\\det
\n \\begin{pmatrix}
\n a_{11}&a_{12}&\\cdots&a_{1n}\\\\
\n \\vdots&\\vdots&&\\vdots\\\\
\n a_{i1}’&a_{i2}’&\\cdots&a_{in}’\\\\
\n \\vdots&\\vdots&&\\vdots\\\\
\n a_{n1}&a_{n2}&\\cdots&a_{nn}
\n \\end{pmatrix}\\\\
\n\t\t &=\\sum_{i=1}^n\\det
\n \\begin{pmatrix}
\n a_{11}&a_{12}&\\cdots&a_{1n}\\\\
\n \\vdots&\\vdots&&\\vdots\\\\
\n \\sum_{k=1}^nx_{ik}a_{k1}&\\sum_{k=1}^nx_{ik}a_{k2}&\\cdots&\\sum_{k=1}^nx_{ik}a_{kn}\\\\
\n \\vdots&\\vdots&&\\vdots\\\\
\n a_{n1}&a_{n2}&\\cdots&a_{nn}
\n \\end{pmatrix}\\\\
\n\t\t &=\\sum_{i=1}^n\\sum_{k=1}^n x_{ik}\\det
\n \\begin{pmatrix}
\n a_{11}&a_{12}&\\cdots&a_{1n}\\\\
\n \\vdots&\\vdots&&\\vdots\\\\
\n a_{k1}&a_{k2}&\\cdots&a_{kn}\\\\
\n \\vdots&\\vdots&&\\vdots\\\\
\n a_{n1}&a_{n2}&\\cdots&a_{nn}
\n \\end{pmatrix}\\\\
\n\t\t &=\\sum_{i=1}^nx_{ii}\\det
\n \\begin{pmatrix}
\n a_{11}&a_{12}&\\cdots&a_{1n}\\\\
\n \\vdots&\\vdots&&\\vdots\\\\
\n a_{i1}&a_{i2}&\\cdots&a_{in}\\\\
\n \\vdots&\\vdots&&\\vdots\\\\
\n a_{n1}&a_{n2}&\\cdots&a_{nn}
\n \\end{pmatrix}=\\mathrm{tr}(X)\\det A.
\n \\end{aligned}
\n \\]
\n<\/div>
\n
\u4f8b\u5b50 1<\/span>.<\/span> \u56de\u5fc6\u4e00\u822c\u7ebf\u6027\u7fa4$\\mathrm{GL}(n, \\mathbb{C})$\u8868\u793a\u590d\u6570\u57df$\\mathbb{C}$\u4e0a$n$\u9636\u53ef\u9006\u65b9\u9635. \u5b50\u7fa4\u7279\u6b8a\u7ebf\u6027\u7fa4$\\mathbb{SL}(n, \\mathbb{C})\\subset \\mathrm{GL}(n, \\mathbb{C})$\u662f\u6ee1\u8db3$A\\in \\mathrm{GL}(n, \\mathbb{C})$, \u4e14$\\det A=1$\u7684$n$\u9636\u65b9\u9635. <\/p>\n

\u7531\u4e8e\u4e00\u4e2a\u77e9\u9635\u7fa4\u7684\u674e\u4ee3\u6570\u5b9a\u4e49\u4e3a\u6052\u7b49\u77e9\u9635\u9644\u8fd1\u7684\u5207\u5411\u91cf\u6784\u6210\u7684\u7ebf\u6027\u7a7a\u95f4(\u5728\u674e\u62ec\u53f7\u79ef\u4f5c\u4e3a\u4e58\u6cd5\u4e0b\u6784\u6210\u4ee3\u6570), \u4ece\u800c\u7279\u6b8a\u7ebf\u6027\u7fa4$\\mathrm{SL}(n, \\mathbb{C})$\u7684\u674e\u4ee3\u6570\u4e3a
\n \\[
\n 0=(\\det 1)’=(\\det(A(t)))’|_{t=0}=\\mathrm{tr}(X)\\det A(0)=\\mathrm{tr}(X).
\n \\]
\n \u5373\u6240\u6709\u7684\u8ff9\u4e3a\u96f6\u7684$n$\u9636\u65b9\u9635.
\n<\/div>
\n\u4e8b\u5b9e\u4e0a, \u4e0a\u9762\u7684\u4f8b\u5b50\u5e76\u6ca1\u6709\u8bf4\u660e\u4e3a\u4ec0\u4e48$\\mathrm{SL}(n, \\mathbb{C})$\u7684\u5207\u7a7a\u95f4\u4e2d\u7684\u5143\u7d20$X$, \u6ee1\u8db3\u65b9\u7a0b$A’=XA$. \u6211\u4eec\u4e0b\u9762\u6765\u89e3\u91ca\u8fd9\u4e00\u70b9.<\/p>\n

\u5c06$\\mathbb{SL}(n, \\mathbb{C})$\u4e2d\u7684\u77e9\u9635$A=(a_{ij})_{n\\times n}$\u89c6\u4e3a$\\mathbb{C}^{n\\times n}$\u4e2d\u7684\u4e00\u4e2a\u5411\u91cf(\u6309\u7167\u9010\u884c\u987a\u6b21\u5199\u51fa\u6784\u6210\u4e00\u4e2a\u5411\u91cf). \u4ece\u800c\u53ef\u4ee5\u5229\u7528 Hermitian \u5185\u79ef\u5b9a\u4e49$\\mathrm{SL}(n, \\mathbb{C})$\u4e0a\u7684\u4e00\u4e2a\u5ea6\u91cf, \u4f7f\u5f97\u5b83\u662f\u4e00\u4e2a\u5149\u6ed1(\u590d)\u6d41\u5f62. \u56e0\u6b64, \u5229\u7528\u6307\u6570\u6620\u5c04\u6211\u4eec\u5f97\u5230
\n\\[
\n A(t)=\\exp(tX(0))A(0)\\implies A'(0)=(\\exp(tX(0))’A(0)=X(0)A(0).
\n\\]
\n\u7531\u57fa\u70b9\u7684\u4efb\u610f\u6027, \u6211\u4eec\u5f97\u5230\u4e00\u822c\u5730$A'(t)=X(t)A(t)$.<\/p>\n","protected":false},"excerpt":{"rendered":"

\u5b9a\u7406 1. \u5047\u8bbe$A(t)$\u662f$n\\times n$\u7684\u5355\u53c2\u6570$n$\u9636\u65b9\u9635, $t\\in(-\\epsilon,\\… \u7ee7\u7eed\u9605\u8bfbLiouville\u5b9a\u7406\u4e0e\u7279\u6b8a\u7ebf\u6027\u7fa4\u7684\u674e\u4ee3\u6570<\/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":[15,147],"class_list":["post-803","post","type-post","status-publish","format-standard","hentry","category-math","tag-liouville","tag-teshuxianxingqun","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/803","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=803"}],"version-history":[{"count":11,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/803\/revisions"}],"predecessor-version":[{"id":814,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/803\/revisions\/814"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=803"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=803"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=803"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}