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Jochemkoenig (talk | contribs) I deleted 'generalized' in section Properties: If a nonsingular matrix and its inverse are both nonnegative matrices (i.e. matrices with nonnegative entries), then the matrix is a permutation matrix. |
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The abstract group of generalized permutation matrices is the [[wreath product]] of ''F''<sup>×</sup> and ''S''<sub>''n''</sub>. Concretely, this means that it is the [[semidirect product]] of Δ(''n'', ''F'') by the [[symmetric group]] ''S''<sub>''n''</sub>:
:Δ(''n'', ''F'')
where ''S''<sub>''n''</sub> acts by permuting coordinates and the diagonal matrices Δ(''n'', ''F'') are isomorphic to the ''n''-fold product (''F''<sup>×</sup>)<sup>''n''</sup>.
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