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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


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

  1. PσPπ=Pσπ for any two permutations σ and π on n objects.
  2. P(1) is the identity matrix.
  3. Permutation matrices are orthogonal matrices and (Pσ)-1=P-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.