# Definition:Permutation Matrix

## Definition

A permutation matrix (of order $n$) is an $n \times n$ square matrix with:

exactly one instance of $1$ in each row and column
$0$ elsewhere.

## Examples

### Full Rook Matrix

An $8 \times 8$ permutation matrix is known as a full rook matrix.

For example:

$\mathbf A = \begin{bmatrix} 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{bmatrix}$

That is, it is a rook matrix in which each row and column has a $1$ in it.

## Also see

• Results about permutation matrices can be found here.