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 correponding 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:

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

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Permutation matrix