Generalized permutation matrix: Difference between revisions

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In [[matrix theorymathematics]], a '''generalized permutation matrix''' (or '''monomial 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. AAn moreexample formal way to express this property is as follows: a [[nonsingular]] matrix ''A'' isof a generalized permutation matrix iff ''A'' can be written as a productis

:<math>\begin{bmatrix}0 & 0 & 3 & 0\\ 0 & -2 & 0 & 0\\

0 & 0 & 3 & 0\\

0 & -2 & 0 & 0\\

1 & 0 & 0 & 0\\

1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}</math>

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'':

:<math> A=DP </math>

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.

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]].

==Group theory==

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>

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 nonsingular diagonal matrices is&Delta;(''n'', ''F'') forms a [[normal subgroup]]. One can show that the group of ''n''&times;''n'' generalized permutation matrices is a [[semidirect product]] of &Delta;(''n'', ''F'') by the [[symmetric group]] ''S''_''n'':

An interesting theorem states the following:

:&Delta;(''n'', ''F'') &#x22CA; ''S''_''n''.

: 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.

Since &Delta;(''n'', ''F'') is isomorphic to (''F''^&times;)^''n'' and ''S''_''n'' acts by permuting coordinates, this group is actually the [[wreath product]] of ''F''^&times; and ''S''_''n''.

==Applications==

Generalized permutationMonomial matrices occur in [[representation theory]] in the context of [[monomial representations]]. A monomial representation of a group ''G'' is a linear representation <math>\, rho\rho:colon G \rightarrow \mathrm{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 permutationmonomial matrices.

[[Category:Matrices]]