## 每週問題 November 14, 2016

Let $m\ge n$ and let $A$ be a complex $m\times n$ matrix of rank $n$. Show that the Hermitian matrix $B=I_m-A(A^{\ast}A)^{-1}A^\ast$ is positive semidefinite.

\begin{aligned} \hbox{trace}B&=\hbox{trace}(I_m-A(A^{\ast}A)^{-1}A^\ast)\\ &=\hbox{trace}{I_m}-\hbox{trace}(A(A^{\ast}A)^{-1}A^\ast)\\ &=\hbox{trace}I_m-\hbox{trace}((A^{\ast}A)^{-1}A^\ast A)\\ &=\hbox{trace}I_m-\hbox{trace}I_n\\ &=m-n, \end{aligned}

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