Info Catalog octave: Expressions Involving Diagonal Matrices octave: Matrix Algebra

octave: Expressions Involving Permutation Matrices

 
 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.
 

Info Catalog octave: Expressions Involving Diagonal Matrices octave: Matrix Algebra