## 每週問題 April 17, 2017

Let $A$ and $B$ be $n\times n$ matrices. If $A^2B+BA^2=2ABA$, show that $(AB-BA)^n=0$.

## 每週問題 April 10, 2017

Let $\mathbf{x}$ and $\mathbf{y}$ be $n$-dimensional column vectors. Prove that

$\hbox{adj}(I-\mathbf{x}\mathbf{y}^T)=\mathbf{x}\mathbf{y}^T+(1-\mathbf{y}^T\mathbf{x})I$.

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## 每週問題 April 3, 2017

Find all matrices commuting with $E$, where $E$ is the matrix all elements of which are equal to $1$.

## 每週問題 March 27, 2017

Prove the following statements.
(a) Let $D=\hbox{diag}(d_1,\ldots,d_n)$, where $d_i$ are distinct. If $AD=DA$, then $A$ is a diagonal matrix.
(b) Let $D=\hbox{diag}(d_1,\ldots,d_n)$, where $d_i$ are distinct and nonzero, and $N=[n_{ij}]$ be an $n\times n$ matrix, where $n_{ij}=\delta_{i+1,j}$. If $AD=DA$ and $NDA=AND$, then $A=aI$.

## 每週問題 March 20, 2017

Let $A$ and $B$ be normal matrices such that $C(A)\perp C(B)$, where $C(X)$ denotes the column space of $X$. Prove that $A+B$ is a normal matrix. Note that $P$ is a normal matrix if $P^\ast P=PP^\ast$.

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## 每週問題 March 13, 2017

Let $A$ be a normal matrix, i.e., $A^\ast A=AA^\ast$. Prove that there exists a normal matrix $B$ such that $A=B^2$.

## 每週問題 March 6, 2017

Let $A$ be an $n\times n$ nonzero Hermitian matrix. Prove that

$\displaystyle \hbox{rank}A\ge\frac{(\hbox{trace}A)^2}{\hbox{trace}(A^2)}$.

## 每週問題 February 27, 2017

Let $A$ and $B$ be Hermitian matrices. We will write that $A\succ B$ if $A-B$ is positive definite. The inequality $A\succ 0$ means that $A$ is positive definite. Prove that if $A\succ B\succ 0$, then $B^{-1}\succ A^{-1}$.

## 每週問題 February 20, 2017

Let $A$ be a $3\times 3$ real orthogonal matrix and $\det A=1$. Prove that

$(\hbox{trace}A)^2-\hbox{trace}(A^2)=2\,\hbox{trace}A$.

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## 每週問題 February 13, 2017

Let $A$ be a unitary matrix, i.e., $A^\ast=A^{-1}$. Prove that

$\displaystyle \lim_{m\to\infty}\frac{1}{m}\sum_{k=0}^{m-1}A^k\mathbf{x}=P\mathbf{x}$,

where $P$ is the Hermitian projection matrix onto $N(I-A^\ast)$.

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