## 每週問題 November 3, 2014

Let $A$ be an $n\times n$ real matrix. Prove the following statements.

(a) If $A$ is symmetric, then $A^2$ is positive semidefinite.
(b) If $A$ is skew-symmetric, then $-A^2$ is positive semidefinite.

(a) 若 $A^T=A$，對於任意 $\mathbf{x}\in\mathbb{R}^n$

$\displaystyle \mathbf{x}^TA^2\mathbf{x}=\mathbf{x}^T(A^TA)\mathbf{x}=(A\mathbf{x})^T(A\mathbf{x})=\Vert A\mathbf{x}\Vert^2\ge 0$

(b) 若 $A^T=-A$，對於任意 $\mathbf{x}\in\mathbb{R}^n$

$\displaystyle \mathbf{x}^T(-A^2)\mathbf{x}=\mathbf{x}^T(A^TA)\mathbf{x}=(A\mathbf{x})^T(A\mathbf{x})=\Vert A\mathbf{x}\Vert^2\ge 0$