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

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

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.

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.

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.