## 每週問題 December 26, 2016

Let $A$ be an $n\times n$ real matrix. Show that $A$ and $A^{-1}$ have all elements nonnegative if and only if each row and each column of $A$ has exactly one positive element and the rest of the elements are zeros.

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## 約束最小平方問題

$A$ 為一個 $m\times n$ 階實矩陣，$\mathbf{b}\in\mathbb{R}^m$。如果線性方程 $A\mathbf{x}=\mathbf{b}$ 是不一致的 (即不存在解)，實務的作法是將線性方程問題改為最小平方近似問題：

$\displaystyle \min_{\mathbf{x}}\Vert A\mathbf{x}-\mathbf{b}\Vert^2$

$\displaystyle \hat{\mathbf{x}}=(A^TA)^{-1}A^T\mathbf{b}$

$\displaystyle \min_{C\mathbf{x}=\mathbf{d}}\Vert A\mathbf{x}-\mathbf{b}\Vert^2$

$\displaystyle \min_{\Vert\mathbf{x}\Vert=d}\Vert A\mathbf{x}-\mathbf{b}\Vert^2$

## 每週問題 December 19, 2016

Let $\mathcal{V}$ be a vector space, $\dim\mathcal{V}=n$, and let $\mathbf{x}_1,\ldots,\mathbf{x}_m\in\mathcal{V}$. Prove that if $m\ge n+2$, then there exist scalars $c_1,\ldots,c_m$ not all of them equal to zero such that $\sum_{i=1}^mc_i\mathbf{x}_i=\mathbf{0}$ and $\sum_{i=1}^mc_i=0$.

## 每週問題 December 12 , 2016

An $n\times n$ matrix is said to be a semi-magic matrix if the sums of the rows and columns are all equal. Show that a semi-magic matrix $A$ can be decomposed as $A=B+C$ such that for integer $k\ge 1$,

$A^k=B^k+C^k$.

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## 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.