## 每週問題 July 20, 2015

Let $A$ and $B$ be $n\times n$ matrices, $n\ge 2$. We say that $A$ and $B$ are equivalent if there exist nonsingular matrices $P$ and $Q$ such that $B=PAQ$. Show that every $n\times n$ matrix $M$ is equivalent to a matrix $D$ where all diagonal elements are zero.

$D=\begin{bmatrix} I_r&0\\ 0&0 \end{bmatrix}\begin{bmatrix} 0&1&0&\cdots&0\\ 0&0&1&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\\ 0&0&0&\cdots&1\\ 1&0&0&\cdots&0 \end{bmatrix}$

This entry was posted in pow 線性方程與矩陣代數, 每週問題 and tagged . Bookmark the permalink.