## 每週問題 May 9, 2016

$A$ 是一個可逆非負矩陣且 $A^{-1}$ 也是非負矩陣，則 $A$ 有甚麼特殊性質？

Let $A=[a_{ij}]$ be an $n\times n$ non-negative matrix, i.e., $a_{ij}\ge 0$ for all $i$ and $j$. If $A$ is nonsingular and $A^{-1}$ is a non-negative matrix as well, then $A$ is a monomial matrix. Note that a nonsingular matrix $A$ is monomial if $A$ can be written as $A=PD$, where $P$ is a permutation matrix and $D$ is a nonsingular diagonal matrix.

$\displaystyle 0=\delta_{ij}=\sum_{l=1}^na_{il}b_{lj}\ge a_{ik}b_{kj}>0$