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

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

Structure

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:

$A=DP.$

Group structure

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.

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:

Δ(n, F) ⋉ 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.

Properties

• 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.

Signed permutation group

Main article: Hyperoctahedral group

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

Properties

• It is the Coxeter group $B_{n}$, and has order $2^{n}n!$.
• It is the symmetry group of the hypercube and (dually) of the cross-polytope.
• Its index 2 subgroup of matrices with determinant 1 is the Coxeter group $D_{n}$ and is the symmetry group of the demihypercube.
• It is a subgroup of the orthogonal group.

Applications

Monomial representations

Main article: Monomial representation

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.