## 每週問題 February 25, 2013

Let $A$ be an $n\times n$ real symmetric matrix. Show that $A$ is positive semidefinite if and only if $X^TAX$ is positive semidefinite for all $n\times m$ real matrix $X$.

$\mathbf{y}^T(X^TAX)\mathbf{y}=(X\mathbf{y})^TA(X\mathbf{y})\ge 0$

$X=\begin{bmatrix} \mathbf{x}&\mathbf{0}&\cdots&\mathbf{0} \end{bmatrix}$

$X^TAX=\begin{bmatrix} \mathbf{x}^T\\ \mathbf{0}^T\\ \vdots\\ \mathbf{0}^T \end{bmatrix}A\begin{bmatrix} \mathbf{x}&\mathbf{0}&\cdots&\mathbf{0} \end{bmatrix}=\begin{bmatrix} \mathbf{x}^TA\mathbf{x}&0&\cdots&0\\ 0&0&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&0 \end{bmatrix}$

PowSol-Feb-25-13