## 每週問題 December 5, 2016

Let $A$ be any $n\times n$ complex matrix. Show that for each positive integer $k$ there exists a unique matrix $B$ such that $A=B(B^\ast B)^k$.

## 每週問題 November 28, 2016

Let $A=[a_{ij}]$ be an $n\times n$ Hermitian and positive semidefinite matrix and $B=[b_{ij}]$ with the property $b_{ij}=1/a_{ij}$. Show that $B$ is positive semidefinite if and only if $\hbox{rank}A=1$.

## 每週問題 November 21, 2016

Let $A$ and $B$ be $n\times n$ Hermitian and positive semidefinite matrices. Show that

$\det(A+B)\ge \det A+\det B$.

## 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.

## 每週問題 November 7, 2016

Let $A=[a_{ij}]$ be an $n\times n$ Hermitian matrix whose eigenvalues, including multiple appearances, are the diagonal elements $a_{ii}$, $i=1,\ldots,n$. Prove that $A$ is diagonal.

## 每週問題 October 31, 2016

Let $A$ and $B$ be $n \times n$ idempotent matrices, i.e., $A^2 = A$ and $B^2 = B$. Show that $A - B$ is idempotent if and only if $AB = BA = B$.

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## 每週問題 October 24, 2016

Let $A$ be an $m\times n$ complex matrix. Prove or disprove the following statements.

(a) $\hbox{rank}(A^\ast A)=\hbox{rank}A$.
(b) $\hbox{rank}(A^TA)=\hbox{rank}A$.

## 每週問題 October 17, 2016

Let $\mathbf{a}_1,\ldots,\mathbf{a}_n$ be $n$ vectors in $\mathbb{R}^m$. Show that $m\times m$ Gramian matrix $G=\sum_{i=1}^n\mathbf{a}_i\mathbf{a}_i^T$ is nonsingular if and only if $\hbox{span}\{\mathbf{a}_1,\ldots,\mathbf{a}_n\}=\mathbb{R}^m$.

## 每週問題 October 10, 2016

Let $A=[a_{ij}]$ be an $n\times n$ Hermitian matrix. If $A$ is positive semidefinite and $a_{ii}=0$ for some $i$, show that $a_{ij}=a_{ji}=0$ for all $j$.