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

• Definition:Group of Permutation Matrices
• Definition:Permutation Matrix Corresponding to Permutation

• Results about permutation matrices can be found here.

Sources

• 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.2$: Functions on vectors: $\S 2.2.3$: $m \times n$ matrices
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): permutation matrix