21.2.2 Expressions Involving Permutation Matrices

If P is a permutation matrix and M a matrix, the expression P*M will permute the rows of M. Similarly, M*P will yield a column permutation. Matrix division P\M and M/P can be used to do inverse permutation.

The previously described syntax for creating permutation matrices can actually help an user to understand the connection between a permutation matrix and a permuting vector. Namely, the following holds, where I = eye (n) is an identity matrix:

  I(p,:) * M = (I*M) (p,:) = M(p,:)

Similarly,

  M * I(:,p) = (M*I) (:,p) = M(:,p)

The expressions I(p,:) and I(:,p) are permutation matrices.

A permutation matrix can be transposed (or conjugate-transposed, which is the same, because a permutation matrix is never complex), inverting the permutation, or equivalently, turning a row-permutation matrix into a column-permutation one. For permutation matrices, transpose is equivalent to inversion, thus P\M is equivalent to P'*M. Transpose of a permutation matrix (or inverse) is a constant-time operation, flipping only a flag internally, and thus the choice between the two above equivalent expressions for inverse permuting is completely up to the user’s taste.

Multiplication and division by permutation matrices works efficiently also when combined with sparse matrices, i.e., P*S, where P is a permutation matrix and S is a sparse matrix permutes the rows of the sparse matrix and returns a sparse matrix. The expressions S*P, P\S, S/P work analogically.

Two permutation matrices can be multiplied or divided (if their sizes match), performing a composition of permutations. Also a permutation matrix can be indexed by a permutation vector (or two vectors), giving again a permutation matrix. Any other operations do not generally yield a permutation matrix and will thus trigger the implicit conversion.