## 每週問題 November 25, 2013

Let $A$ be an $n\times n$ real matrix, and let $A=U\Sigma V^T$ be a singular value decomposition of $A$. Note that $U$ and $V$ are $n\times n$ real orthogonal matrices, and $\Sigma=\hbox{diag}(\sigma_1,\ldots,\sigma_n)$, where $\{\sigma_i\}$ is the set of singular values of $A$. Let $Q$ be an $n\times n$ real orthogonal matrix. Show that $\hbox{trace}(AQ)$ is maximized and $AQ$ is a real symmetric positive semidefinite matrix if $Q=VU^T$.

$\displaystyle \hbox{trace}(AQ)=\hbox{trace}(U\Sigma V^TQ)=\hbox{trace}(\Sigma V^TQU)=\sum_{i=1}^n\sigma_i(V^TQU)_{ii}$