Chemistry - Generalized permutation matrix

In matrix theory, a generalized permutation 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. A more formal way to express this property is as follows: a nonsingular matrix A is a generalized permutation matrix iff A can be written as a product

A = D P

where D is a nonsingular diagonal matrix and P is a permutation matrix. 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 diagonal matrices is a normal subgroup.

An example of a generalized permutation matrix is

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

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.

Applications

Generalized permutation matrices occur in representation theory in the context of monomial representations . A monomial representation of a group G is a linear representation $\, \rho: G \rightarrow GL(n,F)$ of G (here F is the defining field of the representation) such that the image $ρ(G)$ is a subgroup of the group of generalized permutation matrices.

Categories: Matrices