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In [[matrix theory]], a '''generalized permutation matrix''' is a [[matrix (mathematics)|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
:<math> A=DP </math>
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
:<math>\begin{bmatrix}0 & 0 & 3 & 0\\ 0 & -2 & 0 & 0\\
1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}</math>
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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 <math>\, \rho: G \rightarrow GL(n,F) </math> of G (here ''F'' is the defining field of the representation) such that the image <math> \rho(G) </math> is a subgroup of the group of generalized permutation matrices.
[[Category:Matrices]]
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