\u5728\u7ebf\u6027\u4ee3\u6570\u4e2d, \u4e00\u4e2a\u719f\u77e5\u7684\u4e8b\u5b9e\u662f\u4e00\u4e2a\u5b9e\u5bf9\u79f0\u9635\u548c\u4e00\u4e2a\u6b63\u5b9a\u9635\u53ef\u4ee5\u540c\u65f6\u5408\u540c\u4e8e\u5bf9\u89d2\u578b. \u800c\u6700\u8fd1\u6709\u540c\u5b66\u95ee\u6211\u8be5\u7ed3\u8bba\u80fd\u4e0d\u80fd\u63a8\u5e7f\u4e3a\u4e24\u4e2a\u5b9e\u5bf9\u79f0\u9635\u53ef\u4ee5\u540c\u65f6\u5408\u540c\u4e8e\u5bf9\u89d2\u578b?<\/p>\n
\u7a0d\u52a0\u5206\u6790\u53ef\u77e5, \u8fd9\u4e2a\u95ee\u9898\u4e3e\u53cd\u4f8b\u4e0d\u662f\u90a3\u4e48\u5bb9\u6613, \u4f46\u662f\u6211\u5728\u90ed\u8001\u5e08\u7684\u7ebf\u6027\u4ee3\u6570[1]<\/a>\u4e66\u4e0a\u770b\u5230\u5982\u4e0b\u53cd\u4f8b: \u5728\u7ebf\u6027\u4ee3\u6570\u4e2d, \u4e00\u4e2a\u719f\u77e5\u7684\u4e8b\u5b9e\u662f\u4e00\u4e2a\u5b9e\u5bf9\u79f0\u9635\u548c\u4e00\u4e2a\u6b63\u5b9a\u9635\u53ef\u4ee5\u540c\u65f6\u5408\u540c\u4e8e\u5bf9\u89d2\u578b. …<\/p>\n
\n\u5047\u8bbe$A=\\pmatrix{1&0\\\\0&-1}$, $B=\\pmatrix{0&1\\\\1&0}$. \u660e\u663e\u4ed6\u4eec\u90fd\u662f\u5b9e\u5bf9\u79f0\u7684.
\n
\n\u73b0\u5728, \u5982\u679c\u5b58\u5728\u53ef\u9006\u77e9\u9635<\/strong>$P$, \u4f7f\u5f97$PAP^T=\\pmatrix{\\lambda_1&0\\\\0&\\lambda_2}$, $PBP^T=\\pmatrix{\\mu_1&0\\\\0&\\mu_2}$, \u90a3\u4e48\u7531\u4e8e$B$\u7684\u79e9\u548c$PBP^T$\u7684\u79e9\u4e00\u6837, \u6211\u4eec\u77e5\u9053$\\mu_1,\\mu_2$\u90fd\u4e0d\u4e3a\u96f6. \u8fd9\u6837, \u8003\u5bdf\u591a\u9879\u5f0f:
\n\\[
\n(\\lambda_1-\\mu_1 x)(\\lambda_2-\\mu_2 x)=\\det\\left(
\n\\pmatrix{\\lambda_1&0\\\\0&\\lambda_2}-x\\pmatrix{\\mu_1&0\\\\0&\\mu_2}
\n\\right)
\n\\]
\n\u5b83\u6709\u4e24\u4e2a\u5b9e\u6839. \u4f46\u662f, \u6ce8\u610f\u5230
\n\\begin{align*}
\n\\det\\left(
\n\\pmatrix{\\lambda_1&0\\\\0&\\lambda_2}-x\\pmatrix{\\mu_1&0\\\\0&\\mu_2}\\right)
\n&=\\det\\left(
\nP(A-xB)P^T
\n\\right)\\\\
\n&=(\\det P)^2\\det\\pmatrix{1&-x\\\\-x&-1}\\\\
\n&=-(\\det P)^2(1+x^2),
\n\\end{align*}
\n\u5b83\u6ca1\u6709\u5b9e\u6839. \u8fd9\u5c31\u5f97\u5230\u4e86\u77db\u76fe! <\/p>\nReference<\/h4>\n
\n