Generalized permutation matrix: Difference between revisions

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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 ''&times ;''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

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==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]]