Permutation matrix/Definition

For a permutation ${\displaystyle {}\pi }$ on ${\displaystyle {}\{1,\ldots ,n\}}$, the ${\displaystyle {}n\times n}$-matrix

${\displaystyle {}M_{\pi }={\left(a_{ij}\right)}\,,}$

where

${\displaystyle {}a_{\pi (j),j}=1\,}$

and all other entries are ${\displaystyle {}0}$, is called a permutation matrix.