GENERALIZED PERMUTATION MATRIX

In mathematics, a 'generalized permutation matrix' (or 'monomial matrix') is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. An example of a generalized permutation matrix is
:
0 & 0 & 3 & 0\
0 & -2 & 0 & 0\
1 & 0 & 0 & 0\
0 & 0 & 0 & 1end{bmatrix}
A nonsingular matrix ''A'' is a generalized permutation matrix if and only if it can be written as a product of a nonsingular diagonal matrix ''D'' and a permutation matrix ''P'':
:$A=DP$
An interesting theorem states the following: If a nonsingular matrix and its inverse are both nonnegative matrices (i.e. matrices with nonnegative entries), then the matrix is a generalized permutation matrix.
A 'signed permutation matrix' is a generalized permutation matrix whose nonzero entries are ±1.

Contents

Group theory

Applications

Group theory

The set of ''n''×''n'' generalized permutation matrices with entries in a field ''F'' forms a subgroup of the general linear group GL(''n'',''F''), in which the group of nonsingular diagonal matrices Δ(''n'', ''F'') forms a normal subgroup. One can show that the group of ''n''×''n'' generalized permutation matrices is a semidirect product of Δ(''n'', ''F'') by the symmetric group ''S''_''n'':
:Δ(''n'', ''F'') ''S''_''n''.
Since Δ(''n'', ''F'') is isomorphic to (''F''^×)^''n'' and ''S''_''n'' acts by permuting coordinates, this group is actually the wreath product of ''F''^× and ''S''_''n''.

Applications

Monomial matrices occur in representation theory in the context of monomial representations. A monomial representation of a group ''G'' is a linear representation ''ρ'' : ''G'' → GL(''n'', ''F'' ) of ''G'' (here ''F'' is the defining field of the representation) such that the image ''ρ''(''G'' ) is a subgroup of the group of monomial matrices.