## 每週問題 October 1, 2012

Let $A$ be an $n\times n$ matrix. Show that $A^\ast A$ and $AA^\ast$ are unitarily similar, i.e., there exists a unitary matrix $Q$ such that

$AA^\ast=QA^\ast AQ^\ast$.

Note that $Q$ is unitary if $QQ^\ast = Q^\ast Q = I$.

$A$ 的奇異值分解為 $A = U\Sigma V^\ast$，其中 $\Sigma$ 為一 $n\times n$ 階實對角矩陣，$U$$V$$n\times n$ 階么正 (unitary) 矩陣。計算

$A^\ast A = V\Sigma U^\ast =U\Sigma V^\ast= V\Sigma^2V^\ast$

$AA^\ast= U\Sigma V^\ast V\Sigma U^\ast= U\Sigma^2 U^\ast=(UV^\ast)(V\Sigma^2V^\ast)VU^\ast=Q(A^\ast A)Q^\ast$

PowSol-Oct-1-12