## 每週問題 July 18, 2016

Prove that each of the following statements is true.
(a) If $A=[a_{ij}]$ is skew symmetric, then $a_{ii}=0$ for each $i$.
(b) If $A=[a_{ij}]$ is skew Hermitian, then each $a_{ii}$ is a pure imaginary number.
(c) If $A$ is real and symmetric, then $B=\mathrm{i}A$ is skew Hermitian, where $\mathrm{i}=\sqrt{-1}$.

(a) 若 $A^T=-A$，則 $a_{ji}=-a_{ij}$。當 $i=j$$a_{ii}=-a_{ii}$，故 $a_{ii}=0$

(b) 若 $A^\ast=-A$，則 $\overline{a_{ji}}=-a_{ij}$。當 $i=j$$\overline{a_{ii}}=-a_{ii}$。寫出 $a_{ii}=a+\mathrm{i}b$，其中 $\mathrm{i}=\sqrt{-1}$，即有 $a-\mathrm{i}b=-a-\mathrm{i}b$，因此 $a=0$，故得證。

(c) 因為 $\overline{A}=A$$A^T=A$，可得

$B^\ast=(\mathrm{i}A)^\ast=-\mathrm{i}A^\ast=-\mathrm{i}\overline{A}^T=-\mathrm{i}A^T=-\mathrm{i}A=-B$