{"id":80,"date":"2024-07-12T23:04:00","date_gmt":"2024-07-12T15:04:00","guid":{"rendered":"http:\/\/www.triode.cc\/?p=80"},"modified":"2025-09-28T09:39:44","modified_gmt":"2025-09-28T01:39:44","slug":"jnf-and-mdlp","status":"publish","type":"post","link":"https:\/\/www.triode.cc\/index.php\/2024\/07\/12\/jnf-and-mdlp\/","title":{"rendered":"\u82e5\u5c14\u5f53\u6807\u51c6\u578b\u4e0e\u77e9\u9635\u79bb\u6563\u5bf9\u6570\u95ee\u9898"},"content":{"rendered":"\n

\u60f3\u5199\u8fd9\u7bc7\u597d\u4e45\u4e86\uff0c\u4e00\u76f4\u6ca1\u7a7a\uff0c\u8003\u5b8c\u671f\u672b\u6c34\u4e00\u4e0b<\/s><\/p>\n\n\n\n

\u7b2c\u4e00\u90e8\u5206\uff1a\u82e5\u5c14\u5f53\u6807\u51c6\u578b\u6982\u8ff0<\/h2>\n\n\n\n
\n

\u53c2\u8003\u8d44\u6599\uff1a\u300a\u9ad8\u7b49\u4ee3\u6570\uff08\u7b2c\u4e94\u7248\uff09\u300b\uff08\u5317\u4eac\u5927\u5b66\u6570\u5b66\u7cfb\u524d\u4ee3\u6570\u5c0f\u7ec4 \u7f16\uff09:P213-P219,P236-P239<\/p>\n<\/blockquote>\n\n\n\n

\u5b9a\u4e49\uff1a<\/strong>\u5f62\u4e3a<\/p>\n\n\n\n

$$\\pmb{J}(\\lambda_0,k)= \\left(\\begin{matrix} \\lambda_0&0&0&\\cdots&0&0&0\\\\ 1&\\lambda_0&0&\\cdots&0&0&0\\\\ 0&1&\\lambda_0&\\cdots&0&0&0\\\\ \\vdots&\\vdots&\\vdots& &\\vdots&\\vdots&\\vdots\\\\ 0&0&0&\\cdots&1&\\lambda_0&0\\\\ 0&0&0&\\cdots&0&1&\\lambda_0 \\end{matrix}\\right)_{k\u00d7k}$$<\/p>\n\n\n\n

\u7684\u77e9\u9635\u79f0\u4e3a\u82e5\u5c14\u5f53\u5757<\/strong>\uff0c\u5176\u4e2d\\(\\lambda_0\\)\u4e3a\u590d\u6570.\u7531\u82e5\u5e72\u4e2a\u82e5\u5c14\u5f53\u5757\u7ec4\u6210\u7684\u51c6\u5bf9\u89d2\u77e9\u9635<\/p>\n\n\n\n

$$\\pmb{A}=\\left( \\begin{matrix} \\pmb{J}(\\lambda_1,k_1)&&&\\\\ &\\pmb{J}(\\lambda_2,k_2)&&\\\\ &&\\ddots&&\\\\ &&&\\pmb{J}(\\lambda_s,k_s) \\end{matrix} \\right)$$<\/p>\n\n\n\n

\u79f0\u4e3a\u82e5\u5c14\u5f53\u5f62\u77e9\u9635<\/strong>\uff0c\u5176\u4e2d\\(\\lambda_1,\\lambda_2,\\cdots,\\lambda_s\\)\u4e3a\u590d\u6570\uff0c\u6709\u4e00\u4e9b\u53ef\u4ee5\u76f8\u540c.<\/p>\n\n\n\n

\u4f8b\u5982\u77e9\u9635<\/p>\n\n\n\n

$$\\pmb{A}=\\left( \\begin{matrix} 1&0&0\\\\ 1&1&0\\\\ 0&1&1 \\end{matrix} \\right)=\\pmb{J}(1,3)$$<\/p>\n\n\n\n

\u4e3a\u82e5\u5c14\u5f53\u5f62\u77e9\u9635\uff08\u540c\u65f6\u5b83\u4e5f\u662f\u4e00\u4e2a\u82e5\u5c14\u5f53\u5757\uff09.<\/p>\n\n\n\n

\u4e0b\u9762\u7ed9\u51fa\u7684\u662f\u8ddf\u82e5\u5c14\u5f53\u5f62\u77e9\u9635\u76f8\u5173\u7684\u4e00\u4e2a\u91cd\u8981\u7ed3\u8bba\uff0c\u4e5f\u662f\u540e\u9762\u901a\u8fc7\u82e5\u5c14\u5f53\u5f62\u77e9\u9635\u89e3\u51b3\u77e9\u9635\u79bb\u6563\u5bf9\u6570\u95ee\u9898\u7684\u91cd\u8981\u7406\u8bba\u652f\u6491.<\/p>\n\n\n\n

\u5b9a\u7406\uff1a<\/strong>\u6bcf\u4e2a\\(n\\)\u9636\u590d\u77e9\u9635\\(\\pmb{A}\\)\u4e00\u5b9a\u4e0e\u4e00\u4e2a\u82e5\u5c14\u5f53\u5f62\u77e9\u9635\u76f8\u4f3c.\u8fd9\u4e2a\u82e5\u5c14\u5f53\u5f62\u77e9\u9635\u9664\u53bb\u5176\u4e2d\u82e5\u5c14\u5f53\u5757\u7684\u6392\u5217\u987a\u5e8f\u5916\u7531\\(\\pmb{A}\\)\u552f\u4e00\u786e\u5b9a\uff0c\u6211\u4eec\u79f0\u8be5\u82e5\u5c14\u5f53\u5f62\u77e9\u9635\u4e3a\\(\\pmb{A}\\)\u7684\u82e5\u5c14\u5f53\u6807\u51c6\u578b<\/strong>.\u800c\u82e5\u5c14\u5f53\u6807\u51c6\u578b\u4e2d\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u5143\u7d20\u4e3a\\(\\pmb{A}\\)\u7684\u6240\u6709\u7279\u5f81\u503c.<\/p>\n\n\n\n

\u7531\u4e0a\u8ff0\u5b9a\u7406\u6211\u4eec\u53ef\u4ee5\u77e5\u9053\uff1a\u5bf9\u4e8e\u590d\u77e9\u9635\\(\\pmb{A}\\)\u7684\u82e5\u5c14\u5f53\u6807\u51c6\u578b\\(\\pmb{J}\\)\uff0c\u5fc5\u5b58\u5728\u4e00\u53ef\u9006\u77e9\u9635\\(\\pmb{P}\\)\uff0c\u4f7f\u5f97\\(\\pmb{J}=\\pmb{P}^{-1}\\pmb{A}\\pmb{P}\\).<\/p>\n\n\n\n

\u5b9e\u9645\u4e0a\uff0c\u5bf9\u4e8e\u4e0a\u9762\u7684\u82e5\u5c14\u5f53\u5f62\u77e9\u9635\uff1a<\/p>\n\n\n\n

$$\\pmb{A}=\\left( \\begin{matrix} \\pmb{J}(\\lambda_1,k_1)&&&\\\\ &\\pmb{J}(\\lambda_2,k_2)&&\\\\ &&\\ddots&&\\\\ &&&\\pmb{J}(\\lambda_s,k_s) \\end{matrix} \\right)$$<\/p>\n\n\n\n

\u5176\u8f6c\u7f6e\\(\\pmb{A}^T\\)\u4e5f\u53ef\u88ab\u79f0\u4e3a\u82e5\u5c14\u5f53\u5f62\u77e9\u9635\uff0c\u4f8b\u5982\u5bf9\u4e8e\u77e9\u9635<\/p>\n\n\n\n

$$\\pmb{A}=\\left( \\begin{matrix} 1&0&0\\\\ 1&1&0\\\\ 0&1&1\\\\ \\end{matrix} \\right)$$<\/p>\n\n\n\n

\u77e9\u9635\\(\\pmb{A}^T=\\left(\\begin{matrix}1&1&0\\\\0&1&1\\\\0&0&1\\end{matrix}\\right)\\)\u4e5f\u662f\u82e5\u5c14\u5f53\u5f62\u77e9\u9635.<\/p>\n\n\n\n

\u4e0b\u9762\u4ecb\u7ecd\u4e00\u79cd\u6c42\u77e9\u9635\u7684\u82e5\u5c14\u5f53\u6807\u51c6\u578b\u7684\u65b9\u6cd5\u2014\u2014\u901a\u8fc7\u77e9\u9635\u7684\u7279\u5f81\u503c\u6c42\u51fa\u5176\u82e5\u5c14\u5f53\u6807\u51c6\u578b\uff1a<\/p>\n\n\n\n

\u7b2c\u4e00\u6b65\uff1a\u5bf9\u4e8e\u4e00\u77e9\u9635\\(\\pmb{A}_{n\\times n}\\)\uff0c\u6c42\u51fa\u5176\u7279\u5f81\u503c\\(\\lambda_1,\\lambda_2,\\cdots,\\lambda_s\\)\uff1b<\/p>\n\n\n\n

\u7b2c\u4e8c\u6b65\uff1a\u6c42\u51fa\u6bcf\u4e2a\u7279\u5f81\u503c\u7684\u51e0\u4f55\u91cd\u6570\uff08\u76f8\u7b49\u7279\u5f81\u503c\u53ea\u9700\u6c42\u4e00\u6b21\uff0c\u8868\u793a\u8be5\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u82e5\u5c14\u5f53\u5757\u7684\u4e2a\u6570\uff09\uff0c\u5176\u4e2d\u5bf9\u4e8e\u7279\u5f81\u503c\\(\\lambda_i\\)\uff0c\u51e0\u4f55\u91cd\u6570\u4e3a\\(n-r(\\lambda_i\\pmb{I}-\\pmb{A})\\)<\/p>\n\n\n\n

\u7b2c\u4e09\u6b65\uff1a\u6c42\u6bcf\u4e2a\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u82e5\u5c14\u5f53\u5757\u7684\u6700\u5927\u9636\u6570\uff0c\u5373\u627e\u5230\u7279\u5f81\u503c\\(\\lambda_i\\)\u5bf9\u5e94\u7684\u4e00\u4e2a\u503c\\(k_i\\)\uff0c\u4f7f\u5f97\\(k_i\\)\u4e3a\u6ee1\u8db3\\(r[(\\lambda_i\\pmb{I}-\\pmb{A})^{k_i}]=r[(\\lambda_i\\pmb{I}-\\pmb{A})^{k_i+1}]\\)\u7684\u6700\u5c0f\u6b63\u6574\u6570<\/p>\n\n\n\n

\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u5bf9\u77e9\u9635\\(\\pmb{A}=\\left(\\begin{matrix}9&0&-36\\\\6&0&-27\\\\0&1&0\\end{matrix}\\right)\\)\uff0c\u6c42\u5176\u5bf9\u5e94\u7684\u82e5\u5c14\u5f53\u6807\u51c6\u578b\uff1a<\/p>\n\n\n\n

\u7b2c\u4e00\u6b65\uff0c\u6c42\u77e9\u9635\\(\\pmb{A}\\)\u5bf9\u5e94\u7684\u7279\u5f81\u503c\uff0c\u5373\u89e3\u65b9\u7a0b\\(\\det(\\lambda \\pmb{I}-A)=0\\)\uff0c\u89e3\u5f97\u552f\u4e00\u89e3\\(\\lambda=3\\)\uff1b<\/p>\n\n\n\n

\u7b2c\u4e8c\u6b65\uff0c\u5bf9\u552f\u4e00\u7279\u5f81\\(\\lambda=3\\)\uff0c\u6709\\(3\\pmb{I}-\\pmb{A}=\\left(\\begin{matrix}-6&0&36\\\\ -6&3&27\\\\0&-1&3\\end{matrix}\\right)\\sim\\left(\\begin{matrix}1&0&-6\\\\0&1&-3\\\\0&0&0\\end{matrix}\\right)\\)\uff0c\u6240\u4ee5\u5176\u51e0\u4f55\u91cd\u6570\u4e3a\\(3-r(3\\pmb{I}-\\pmb{A})=1\\)\uff0c\u6240\u4ee5\u7279\u5f81\u503c\\(\\lambda=3\\)\u5bf9\u5e94\u7684\u82e5\u5c14\u5f53\u5757\u7684\u4e2a\u6570\u4e3a\\(1\\)\uff1b<\/p>\n\n\n\n

\u7b2c\u4e09\u6b65\uff1a\u5bf9\\(3\\pmb{I}-\\pmb{A}\\)\u6709\\(r[(3\\pmb{I}-\\pmb{A})^1]=2,r[(3\\pmb{I}-\\pmb{A})^2]=1,r[(3\\pmb{I}-\\pmb{A})^3]=0,r[(3\\pmb{I}-\\pmb{A})^4]=0\\)\uff0c\u6240\u4ee5\u8be5\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u82e5\u5c14\u5f53\u5757\u7684\u6700\u5927\u9636\u6570\u4e3a\\(3\\)\uff0c\u6240\u4ee5\\(\\lambda=3\\)\u5bf9\u5e94\u7684\u4e00\u4e2a\u4e09\u9636\u7684\u82e5\u5c14\u5f53\u5757\\(\\pmb{J}(3,3)=\\left(\\begin{matrix}3&1&0\\\\0&3&1\\\\0&0&3\\end{matrix}\\right)\\)\u6216\u8005\\(\\pmb{J}(3,3)=\\left(\\begin{matrix}3&0&0\\\\1&3&0\\\\0&1&3\\end{matrix}\\right)\\)\uff0c\u6240\u4ee5\u77e9\u9635\\(\\pmb{A}=\\left(\\begin{matrix}9&0&-36\\\\6&0&-27\\\\0&1&0\\end{matrix}\\right)\\)\u5bf9\u5e94\u7684\u82e5\u5c14\u5f53\u6807\u51c6\u578b\u4e3a\\(\\left(\\begin{matrix}3&1&0\\\\0&3&1\\\\0&0&3\\end{matrix}\\right)\\)\u6216\u8005\\(\\left(\\begin{matrix}3&0&0\\\\1&3&0\\\\0&1&3\\end{matrix}\\right)\\).<\/p>\n\n\n\n

\u7531\u4e8e\u672c\u6587\u4e3b\u8981\u8ba8\u8bba\u7684\u662f\u901a\u8fc7\u82e5\u5c14\u5f53\u6807\u51c6\u578b\u6c42\u89e3\u77e9\u9635\u7684\u79bb\u6563\u5bf9\u6570\u95ee\u9898\uff0c\u6240\u4ee5\u5c31\u4e0d\u518d\u6df1\u5165\u4ecb\u7ecd\u82e5\u5c14\u5f53\u6807\u51c6\u578b\u4e86\u3002<\/p>\n\n\n\n

\u5728\u8fd9\u4e00\u90e8\u5206\u7684\u6700\u540e\uff0c\u6211\u4eec\u4ecb\u7ecdsage\u4e2d\u6c42\u65b9\u9635\u7684\u82e5\u5c14\u5f53\u6807\u51c6\u578b\u7684\u65b9\u6cd5\uff0c\u8fd9\u4e2a\u5728\u540e\u9762\u4e5f\u4f1a\u7528\u5230\uff1a<\/p>\n\n\n\n

\u5728sage\u4e2d\u6709\u51fd\u6570jordan_form()<\/code>\uff0c\u5176\u4f5c\u7528\u662f\u8fd4\u56deself<\/code>\u7684\u82e5\u5c14\u5f53\u6807\u51c6\u578b\uff0c\u5176\u51fd\u6570\u539f\u578b\u4e3ajordan_form(subdivide=True,transformation=False)<\/code>\uff0c\u5176\u63a5\u6536\u4e09\u4e2a\u53c2\u6570\uff0c\u5206\u522b\u4e3a\u4e00\u4e2a\u65b9\u9635self<\/code>\uff0c\u4e00\u4e2a\u5e03\u5c14\u578b\u53d8\u91cfsubdivide<\/code>\uff08\u9ed8\u8ba4\u4e3aTrue<\/code>\uff09\uff0c\u4e00\u4e2a\u5e03\u5c14\u578b\u53d8\u91cftransformation<\/code>\uff08\u9ed8\u8ba4\u4e3aFalse<\/code>\uff09\u3002<\/p>\n\n\n\n

\u82e5subdivide=True<\/code>\uff0c\u5219\u8fd4\u56de\u7684\u77e9\u9635\u4e2d\u4f1a\u6839\u636e\u82e5\u5c14\u5f53\u5757\u8fdb\u884c\u5206\u5757\u6807\u51fa.<\/p>\n\n\n\n

\u82e5transformation=True<\/code>\uff0c\u5219\u4f1a\u8fd4\u56de\u4e00\u4e2a\u77e9\u9635\u7ec4(J,P)<\/code>\uff0c\u5176\u4e2dJ<\/code>\u4e3aself<\/code>\u7684\u82e5\u5c14\u5f53\u6807\u51c6\u578b\uff0c\u800cP<\/code>\u4e3a\u4f7f\u5f97\\(self=\\pmb{P}\\pmb{J}\\pmb{P}^{-1}\\)\u6210\u7acb\u7684\u65b9\u9635\uff1b\u82e5transformation=False<\/code>\uff0c\u5219\u53ea\u4f1a\u8fd4\u56deself<\/code>\u7684\u82e5\u5c14\u5f53\u6807\u51c6\u578bJ<\/code>.<\/p>\n\n\n\n

\u4f8b\uff1a\u6c42\u77e9\u9635\\(\\pmb{A}=\\left(\\begin{matrix}9&0&-36\\\\6&0&-27\\\\0&1&0\\end{matrix}\\right)\\)\u7684\u82e5\u5c14\u5f53\u6807\u51c6\u578b\\(\\pmb{J}\\)\u5e76\u6c42\u77e9\u9635\\(\\pmb{P}\\)\u4f7f\u5f97\\(\\pmb{P}^{-1}\\pmb{A}\\pmb{P}=\\pmb{J}\\)<\/p>\n\n\n\n

sage\u4ee3\u7801\u5982\u4e0b\uff1a<\/p>\n\n\n\n

<\/circle><\/circle><\/circle><\/g><\/svg><\/span>