Generalized permutation matrix

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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

[\begin0 & 0 & 3 & 0\\0 & -2 & 0 & 0\\1 & 0 & 0 & 0\\0 & 0 & 0 & 1\end]

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.

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 [\rho\colon G \rightarrow \mathrm(n,F)] of G (here F is the defining field of the representation) such that the image [ \rho(G) ] is a subgroup of the group of monomial matrices.

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