Definition:Permutation Matrix

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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

$3 \times 3$ Permutation Matrix

Definition:Permutation Matrix/Examples/3 x 3

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.


Sources