# 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. Unlike a permutation matrix, where the nonzero entry must be 1, in a generalized permutation matrix the nonzero entry can be any nonzero value. An example of a generalized permutation matrix is

An invertible matrix *A* is a generalized permutation matrix if and only if it can be written as a product of an invertible diagonal matrix *D* and an (implicitly invertible) permutation matrix *P*: i.e.,

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. Indeed, the generalized permutation matrices are the normalizer of the diagonal matrices, meaning that the generalized permutation matrices are the *largest* subgroup of GL(*n*, *F*) in which diagonal matrices are normal.

The abstract group of generalized permutation matrices is the wreath product of *F*^{×} and *S*_{n}. Concretely, this means that it is the semidirect product of Δ(*n*, *F*) by the symmetric group *S*_{n}:

where *S*_{n} acts by permuting coordinates and the diagonal matrices Δ(*n*, *F*) are isomorphic to the *n*-fold product (*F*^{×})^{n}.

To be precise, the generalized permutation matrices are a (faithful) linear representation of this abstract wreath product: a realization of the abstract group as a subgroup of matrices.

One can generalize further by allowing the entries to lie in a ring, rather than in a field. In that case if the non-zero entries are required to be units in the ring, one again obtains a group. On the other hand, if the non-zero entries are only required to be non-zero, but not necessarily invertible, this set of matrices forms a semigroup instead.

A **signed permutation matrix** is a generalized permutation matrix whose nonzero entries are ±1, and are the integer generalized permutation matrices with integer inverse.

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.