# 一个高等代数问题

[提示: 可以证明, 如果$\sigma\tau=\tau\sigma$, 则$\sigma$与$\tau$至少有一个公共的特征向量.]

V=\oplus_{i=1}^{r}W_{i}=\oplus_{i=1}^{s}M_{i}
\end{align*}

V=\oplus_{i=1}^{r}\oplus_{j=1}^{s}W_{i}\cap M_{j}\tag{1}
\end{align*}

(1)式是个结论，也可以证明一下：

V=\oplus_{i=1}^{r}W_{i}
\end{align*}
(每个$W_{i}$特征值不同)对于任何$A$的不变子空间$W$，我们来证\begin{align*}
W=\oplus_{i=1}^{r}W\cap W_{i}
\end{align*}

\cup_{i=1}^{r}M_{i}\subset W
\end{align*}

\alpha=\sum_{i=1}^{r}\alpha_{i}
\end{align*}

A^k\alpha=\sum_{i=1}^{r}\lambda_{i}^{k}\alpha_{i}(k=1,2,\cdots,r-1)
\end{align*}

\begin{align*}\left( {\begin{array}{*{20}{c}}
\alpha \\
{A\alpha } \\
\vdots \\
{{A^{r - 1}}\alpha } \\
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
1 & 1 & \cdots & 1 \\
{{\lambda _1}} & {{\lambda _2}} & \cdots & {{\lambda _r}} \\
\vdots & \vdots & \ddots & \vdots \\
{\lambda _1^{r - 1}} & {\lambda _2^{r - 1}} & \cdots & {\lambda _r^{r - 1}} \\
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
{{\alpha _1}} \\
{{\alpha _2}} \\
\vdots \\
{{\alpha _r}} \\
\end{array}} \right)\end{align*}

W\subset\cup_{i=1}^{r}M_{i}
\end{align*}

\begin{align*}
W=\oplus_{i=1}^{r}W\cap W_{i}
\end{align*}