(octave.info)Expressions Involving Permutation Matrices

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21.2.2 Expressions Involving Permutation Matrices
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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.