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+又矩阵$A$各行元素之和均为5,所以转换为方程组:\medskip + +$\left\{\begin{array}{l} + A_{11}+A_{12}+A_{13}=5 \\ + A_{21}+A_{22}+A_{23}=5 \\ + A_{31}+A_{32}+A_{33}=5 +\end{array}\right.$,转为矩阵:$\left[\begin{array}{ccc} + A_{11} & A_{12} & A_{13} \\ + A_{21} & A_{22} & A_{23} \\ + A_{31} & A_{32} & A_{33} +\end{array}\right]\left[\begin{array}{c} + 1 \\ + 1 \\ + 1 \\ +\end{array}\right]=5\left[\begin{array}{c} + 1 \\ + 1 \\ + 1 \\ +\end{array}\right]$。\medskip + +即$\xi=(1,1,1)^T$。 + +\subsection{可逆矩阵} + +使用可逆矩阵相似对角化的性质。若$A\sim B$,则$P^{-1}AP=B$。$B$为纯量阵。且$B$的迹为$A$的特征值。$P$为特征向量。\medskip + +\textbf{例题:}已知$P^{-1}AP=\left[\begin{array}{ccc} + 1 \\ + & 1 \\ + & & -1 +\end{array}\right]$,$P=(\alpha_1,\alpha_2,\alpha_3)$可逆,求$A$关于特征值$\lambda=1$的特征向量。 + +解:根据$P^{-1}AP=\Lambda$,所以$P$为特征向量,$1,1,-1$为特征值。 + +\subsection{实对称矩阵} + +实对称矩阵的不同特征值的特征向量相互正交($A^TA=0$)。 + +\textbf{例题:}已知$A$为三阶实对称矩阵,特征值为$1,3,-2$,其中$\alpha_1=(1,2,-2)^T$,$\alpha_2=(4,-1,a)^T$分别属于特征值$\lambda=1$,$\lambda=3$的特征向量。求$A$属于特征值$\lambda=-2$的特征向量。 + \section{相似理论} \section{判断相似对角化}