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Last edited by 青青子衿 at 2019-5-8 14:27:00\begin{align*}
\varphi(\lambda)
&=\det\left(\lambda\boldsymbol{I}_n-\boldsymbol{A}\right)=\sum_{k=0}^{n}c_{\overset{\,}{k}}\lambda^k\\
&=c_{\overset{\,}{n}}\lambda^n+c_{\overset{\,}{n-1}}\lambda^{n-1}+\cdots+c_{\overset{\,}{1}}\lambda+c_{\overset{\,}{0}}\\
&=\lambda^n+c_{\overset{\,}{n-1}}\lambda^{n-1}+\cdots+c_{\overset{\,}{1}}\lambda+c_{\overset{\,}{0}}\\
&=\lambda^n-\operatorname{tr}\left(\boldsymbol{A}\right)\lambda^{n-1}+\cdots+c_{\overset{\,}{1}}\lambda+(-1)^n\det\left(\boldsymbol{A}\right)\\
\end{align*}
\begin{align*}
\sigma_{\overset{\,}{k}}=(-1)^k\,c_{\overset{\,}{n-k}}={\large\dfrac{1}{k!}}\begin{vmatrix}
\operatorname{tr}\left(\boldsymbol{A}\right) & 1 & & & & & \\
\operatorname{tr}\left(\boldsymbol{A}^2\right) & \operatorname{tr}\left(\boldsymbol{A}\right) & 2 & & & & \\
\operatorname{tr}\left(\boldsymbol{A}^3\right) & \operatorname{tr}\left(\boldsymbol{A}^2\right) & \operatorname{tr}\left(\boldsymbol{A}\right) & \ddots & & & \\
\vdots & \vdots & \vdots& \ddots & \ddots & & \\
\operatorname{tr}\left(\boldsymbol{A}^{k-2}\right) & \operatorname{tr}\left(\boldsymbol{A}^{k-3}\right) & \operatorname{tr}\left(\boldsymbol{A}^{k-4}\right) & \dots & \operatorname{tr}\left(\boldsymbol{A}\right) & k-2 & \\
\operatorname{tr}\left(\boldsymbol{A}^{k-1}\right) & \operatorname{tr}\left(\boldsymbol{A}^{k-2}\right) & \operatorname{tr}\left(\boldsymbol{A}^{k-3}\right) & \dots& \operatorname{tr}\left(\boldsymbol{A}^2\right) & \operatorname{tr}\left(\boldsymbol{A}\right) & k-1\\
\operatorname{tr}\left(\boldsymbol{A}^k\right) & \operatorname{tr}\left(\boldsymbol{A}^{k-1}\right) & \operatorname{tr}\left(\boldsymbol{A}^{k-2}\right) & \dots& \operatorname{tr}\left(\boldsymbol{A}^3\right) & \operatorname{tr}\left(\boldsymbol{A}^2\right) & \operatorname{tr}\left(\boldsymbol{A}\right)\\
\end{vmatrix}
\end{align*}
en.wikipedia.org/wiki/Faddeev–LeVerrier_algorithm
\begin{align*}
x_{\overset{\,}{1}}+x_{\overset{\,}{2}}+\cdots+x_{\overset{\,}{n}}=\sigma_{\overset{\,}{1}}
&
=
&
\sum_{i=1}^n\,x_{\overset{\,}{i}}&=-\,c_{\overset{\,}{n-1}}\\
\begin{split}
x_{\overset{\,}{1}}x_{\overset{\,}{2}}+x_{\overset{\,}{1}}x_{\overset{\,}{3}}+&\cdots&+x_{\overset{\,}{1}}x_{\overset{\,}{n}}\\
+x_{\overset{\,}{2}}x_{\overset{\,}{3}}+&\cdots&+x_{\overset{\,}{2}}x_{\overset{\,}{n}}\\
+&\cdots&\\
&&+x_{\overset{\,}{n-1}}x_{\overset{\,}{n}}\\
\end{split}=\sigma_{\overset{\,}{2}}
&
=
&
\sum_{1\leqslant\,i_1<i_2\leqslant\,n}\,x_{\overset{\,}{i_1}}x_{\overset{\,}{i_2}}&=\phantom{+}\,c_{\overset{\,}{n-2}}\\
\\
\left(\begin{array}{cc}
&\begin{split}
x_{\overset{\,}{1}}x_{\overset{\,}{2}}x_{\overset{\,}{3}}+x_{\overset{\,}{1}}x_{\overset{\,}{2}}x_{\overset{\,}{4}}+&\cdots&+x_{\overset{\,}{1}}x_{\overset{\,}{2}}x_{\overset{\,}{n}}\\
+x_{\overset{\,}{1}}x_{\overset{\,}{3}}x_{\overset{\,}{4}}+&\cdots&+x_{\overset{\,}{1}}x_{\overset{\,}{3}}x_{\overset{\,}{n}}\\
+&\cdots&\\
&&+x_{\overset{\,}{1}}x_{\overset{\,}{n-1}}x_{\overset{\,}{n}}
\end{split}\\
&+\\
&\begin{split}
x_{\overset{\,}{2}}x_{\overset{\,}{3}}x_{\overset{\,}{4}}+&\cdots&+x_{\overset{\,}{2}}x_{\overset{\,}{3}}x_{\overset{\,}{n}}\\
+&\cdots&\\
&&+x_{\overset{\,}{2}}x_{\overset{\,}{n-1}}x_{\overset{\,}{n}}\\
\end{split}\\
&+\\
&\vdots\\
&+x_{\overset{\,}{n-2}}x_{\overset{\,}{n-1}}x_{\overset{\,}{n}}
\end{array}\right)
=\sigma_{\overset{\,}{3}}
&
=
&
\sum_{1\leqslant\,i_1<i_2<i_3\leqslant\,n}\,x_{\overset{\,}{i_1}}x_{\overset{\,}{i_2}}x_{\overset{\,}{i_3}}&=-\,c_{\overset{\,}{n-3}}\\
\vdots\,\,\,&&\vdots\\
\sigma_{\overset{\,}{k}}&=&
\sum_{1\leqslant\,i_1<i_2<\cdots<i_k\leqslant\,n}\,x_{\overset{\,}{i_1}}x_{\overset{\,}{i_2}}\cdots\,x_{\overset{\,}{i_k}}&=(-1)^k\,c_{\overset{\,}{n-k}}\\
\vdots\,\,\,&&\vdots\\
\sigma_{\overset{\,}{n}}&=&
x_{\overset{\,}{1}}x_{\overset{\,}{2}}\cdots\,x_{\overset{\,}{n}}&=(-1)^n\,c_{\overset{\,}{0}}\\
\end{align*} |
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