## 每週問題 March 20, 2017

Let $A$ and $B$ be normal matrices such that $C(A)\perp C(B)$, where $C(X)$ denotes the column space of $X$. Prove that $A+B$ is a normal matrix. Note that $P$ is a normal matrix if $P^\ast P=PP^\ast$.

\begin{aligned} (A+B)^\ast(A+B)&=A^\ast A+A^\ast B+B^\ast A+B^\ast B\\ &=A^\ast A+B^\ast B.\end{aligned}

\begin{aligned} (A+B)(A+B)^\ast&=AA^\ast+AB^\ast+BA^\ast+BB^\ast\\ &=AA^\ast+BB^\ast\\ &=A^\ast A+B^\ast B.\end{aligned}

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### 1 Response to 每週問題 March 20, 2017

1. Meiyue Shao says:

用谱分解来证也很方便.