张祖锦常用结论09可交换的复方阵可以同时相似上三角化

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# 张祖锦常用结论可交换的复方阵可以同时相似上三角化 设 $\displaystyle A,B$ 都是 $\displaystyle n$ 阶复矩阵. 证明: 若 $\displaystyle A$ 与 $\displaystyle B$ 乘积可交换, 那么存在 $\displaystyle n$ 阶可逆矩阵 $\displaystyle P$ 使得 $\displaystyle P^\{-1\}AP$ 和 $\displaystyle P^\{-1\}BP$ 都是上三角阵. [纸质资料](https://mp.weixin.qq.com/s/ycnPCSqWFlThEnq9ZZ6gBQ)/[答疑](https://mp.weixin.qq.com/s/JGYZG5rsshf7Z2Amo2di8A)/[pdf1](https://mp.weixin.qq.com/s/Pt6\_h5MqtomrUDYiPEwkxg)/[pdf2](https://mp.weixin.qq.com/s/dWvpeJFKnFr0WYPoidXXMA) / 对 $\displaystyle n$ 作数学归纳法. 当 $\displaystyle n=1$ 时, 结论自明. 假设对阶数 $\displaystyle \leq n-1$ 的复矩阵结论成立, 则对 $\displaystyle n$ 阶复矩阵 $\displaystyle A,B$, 设 $\displaystyle \mathbb\{C\}^n$ 上的线性变换 $\displaystyle \mathscr\{A\}, \mathscr\{B\}$ 在基 $\displaystyle e\_1,\cdots,e\_n$ 下的矩阵分别为 $\displaystyle A,B$, 则 $\displaystyle \mathscr\{A\}\mathscr\{B\}=\mathscr\{B\}\mathscr\{A\}$. 任取 $\displaystyle \mathscr\{A\}$ 的特征值 $\displaystyle \lambda\in\mathbb\{C\}$, 令 \begin\{aligned\} V\_\lambda=\left\\{\alpha\in\mathbb\{C\}^n; \mathscr\{A\}\alpha=\lambda\alpha\right\\}, \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 则由 \begin\{aligned\} \alpha\in V\_\lambda&\Rightarrow \mathscr\{A\}\mathscr\{B\}\alpha=\mathscr\{B\}\mathscr\{A\}\alpha=\mathscr\{B\}(\lambda\alpha) =\lambda \mathscr\{B\}\alpha\\\\ &\Rightarrow \mathscr\{B\}\alpha\in V\_\lambda \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 知 $\displaystyle V\_\lambda$ 上的线性变换 $\displaystyle \mathscr\{B\}|\_\{V\_\lambda\}$ 也有特征值 $\displaystyle \mu$ 及相应的特征向量 $\displaystyle \eta\_1$, 将 $\displaystyle \eta\_1$ 扩充为 $\displaystyle \mathbb\{C\}^n$ 的一组基 $\displaystyle \eta\_1,\cdots,\eta\_n$, 则 \begin\{aligned\} \mathscr\{A\}(\eta\_1,\cdots,\eta\_n)=&(\eta\_1,\cdots,\eta\_n)\left(\begin\{array\}\{cccccccccc\}\lambda&\star \\\\ 0&A\_1\end\{array\}\right),\\\\ \mathscr\{B\}(\eta\_1,\cdots,\eta\_n)=&(\eta\_1,\cdots,\eta\_n)\left(\begin\{array\}\{cccccccccc\}\mu&\star \\\\ 0&B\_1\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 又由 \begin\{aligned\} \mathscr\{A\}(\eta\_1,\cdots,\eta\_n)&=\mathscr\{A\}(e\_1,\cdots,e\_n)(\eta\_1,\cdots\eta\_n)\\\\ &=(e\_1,\cdots,e\_n)A(\eta\_1,\cdots,\eta\_n)\\\\ &\stackrel\{P\_1=(\eta\_1,\cdots,\eta\_n)\}\{=\} (\eta\_1,\cdots,\eta\_n)P\_1^\{-1\}AP\_1. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 于是 \begin\{aligned\} P\_1^\{-1\}AP\_1=\left(\begin\{array\}\{cccccccccc\}\lambda&\star \\\\ 0&A\_1\end\{array\}\right), P\_1^\{-1\}BP\_1=\left(\begin\{array\}\{cccccccccc\}\mu&\star \\\\ 0&B\_1\end\{array\}\right). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 再由 $\displaystyle AB=BA$ 知 $\displaystyle A\_1B\_1=B\_1A\_1$. 而由归纳假设, 存在可逆阵 $\displaystyle P\_2$, 使得 \begin\{aligned\} P\_2^\{-1\}A\_1P\_2=U\_1, P\_2^\{-1\}B\_1P\_2=U\_2 \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 均为上三角阵. 令 \begin\{aligned\} P=P\_1\left(\begin\{array\}\{cccccccccc\}1&\\\\ &P\_2\end\{array\}\right), \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 则 $\displaystyle P$ 可逆, 且 \begin\{aligned\} P^\{-1\}AP=\left(\begin\{array\}\{cccccccccc\}\lambda&\star \\\\ &U\_1\end\{array\}\right), P^\{-1\}BP=\left(\begin\{array\}\{cccccccccc\}\mu&\star \\\\ &U\_2\end\{array\}\right) \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\} 都是上三角阵. 跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw)/[公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow)/