Generalized permutation matrix: Difference between revisions

Revision as of 08:46, 24 November 2009

Revision as of 08:50, 24 November 2009

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:<math> A=DP. </math>

===Group theorystructure===

The set of ''n''&times;''n'' generalized permutation matrices with entries in a [[field (mathematics)|field]] ''F'' forms a [[subgroup]] of the [[general linear group]] GL(''n'',''F''), in which the group of nonsingular diagonal matrices &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'':

The abstract group of generalized permutation matrices is the [[wreath product]] of ''F''^&times; and ''S''_''n''. Concretely, this means that it is the [[semidirect product]] of &Delta;(''n'', ''F'') by the [[symmetric group]] ''S''_''n'':

:&Delta;(''n'', ''F'') {{unicode|&#x22C9;}} ''S''_''n''.,

where ''S''_''n'' acts by permuting coordinates and the diagonal matrices &Delta;(''n'', ''F'') are isomorphic to the ''n''-fold product (''F''^&times;)^''n''.

To be precise, the generalized permutation matrices are a (faithful) [[linear representation]] of this abstract wreath product: a realization of the abstract group as a subgroup of matrices.

== Properties ==

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==

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* Its index 2 subgroup of matrices with determinant 1 is the Coxeter group <math>D_n</math> and is the symmetry group of the [[demihypercube]].

* It is a subgroup of the [[orthogonal group]].

==Group theory==

The set of ''n''&times;''n'' generalized permutation matrices with entries in a [[field (mathematics)|field]] ''F'' forms a [[subgroup]] of the [[general linear group]] GL(''n'',''F''), in which the group of nonsingular diagonal matrices &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'':

:&Delta;(''n'', ''F'') {{unicode|&#x22C9;}} ''S''_''n''.

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==