Generalized permutation matrix: Difference between revisions

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{{Short description|Matrix with one nonzero entry in each row and column}}

In [[mathematics]], 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. Unlike a permutation matrix, where the nonzero entry must be 1, in a generalized permutation matrix the nonzero entry can be any nonzero value. An example of a generalized permutation matrix is

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===Group structure===

The [[set (mathematics)|set]] of ''n'' × ''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 [[invertible matrix|nonsingular]] diagonal matrices Δ(''n'', ''F'') forms a [[normal subgroup]]. Indeed, over all fields except [[GF(2)]], the generalized permutation matrices are the [[normalizer]] of the diagonal matrices, meaning that the generalized permutation matrices are the ''largest'' subgroup of GL(''n'', ''F'') in which diagonal matrices are normal.

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

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== Properties ==

* 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.

:* The determinant of a generalized permutation matrix is given by <math display="block">\det(G)=\det(P)\cdot \det(D)=\operatorname{sgn}(\pi)\cdot d_{11}\cdot \ldots \cdot d_{nn},</math> where <math>\operatorname{sgn}(\pi)</math> is the [[sign of a permutation|sign]] of the [[permutation]] <math>\pi</math> associated with <math>P</math> and <math>d_{11},\ldots ,d_{nn}</math> are the diagonal elements of <math>D</math>.

* The determinant of a generalized permutation matrix is given by

::<math>\det(G)=\det(P)\cdot \det(D)=\operatorname{sgn}(\pi)\cdot d_{11}\cdot \ldots \cdot d_{nn},</math>

: where <math>\operatorname{sgn}(\pi)</math> is the [[sign of a permutation|sign]] of the [[permutation]] <math>\pi</math> associated with <math>P</math> and <math>d_{11},\ldots ,d_{nn}</math> are the diagonal elements of <math>D</math>.

== Generalizations ==

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{{Matrix classes}}

[[Category:Matrices (mathematics)]]

[[Category:Permutations]]

[[Category:Sparse matrices]]