Permutation matrix

In linear algebra, a permutation matrix is a matrix that has exactly one 1 in each row or column and 0s elsewhere. Permutation matrices are the matrix representation of permutations.

For example, the permutation matrix corresponding to σ=(1)(2 4 5 3) is

and

In general, for a permutation σ on n objects, the corresponding permutation matrix is an n-by-n matrix P_σ is given by P_σ[i,j]=1 if i=σ(j) and 0 otherwise. We have

.

Properties:

1. P_σP_π=P_σπ for any two permutations σ and π on n objects.
2. P₍₁₎ is the identity matrix.
3. Permutation matrices are orthogonal matrices and (P_σ)^-1=P_(σ^-1).

See also generalized permutation matrix.

A permutation matrix is a stochastic matrix; in fact doubly stochastic. One can show that all doubly stochastic matrices of a fixed size are convex linear combinations of permutation matrices, giving them a characterisation as the set of extreme points.