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

\begin{bmatrix} 0 & 0 & 3 & 0\\ 0 & -2 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\end{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.

[edit] 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 Sn:

Δ(n, F) ⋉ Sn.

Since Δ(n, F) is isomorphic to (F×)n and Sn acts by permuting coordinates, this group is actually the wreath product of F× and Sn.

[edit] 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.