Inmathematics, ageneralized permutation matrix (ormonomial matrix) is amatrix with the same nonzero pattern as apermutation 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

Aninvertible matrixA is a generalized permutation matrixif and only if it can be written as a product of an invertiblediagonal matrixD and an (implicitlyinvertible)permutation matrixP: i.e.,

${\displaystyle A=DP.}$

Theset ofn ×n generalized permutation matrices with entries in afieldF forms asubgroup of thegeneral linear group GL(n,F), in which the group ofnonsingular diagonal matrices Δ(n,F) forms anormal subgroup. Indeed, over all fields exceptGF(2), the generalized permutation matrices are thenormalizer of the diagonal matrices, meaning that the generalized permutation matrices are thelargest subgroup of GL(n,F) in which diagonal matrices are normal.

The abstract group of generalized permutation matrices is thewreath product ofF^× andS_n. Concretely, this means that it is thesemidirect product of Δ(n,F) by thesymmetric groupS_n:

S_n ⋉ Δ(n,F),

whereS_n acts by permuting coordinates and the diagonal matrices Δ(n,F) areisomorphic to then-fold product (F^×)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.

• The subgroup where all entries are 1 is exactly thepermutation matrices, which is isomorphic to the symmetric group.
• The subgroup where all entries are ±1 is thesigned permutation matrices, which is thehyperoctahedral group.
• The subgroup where the entries aremthroots of unity${\displaystyle {\mu }_m}$ is isomorphic to ageneralized symmetric group.
• The subgroup of diagonal matrices isabelian, normal, and a maximal abelian subgroup. Thequotient group is the symmetric group, and this construction is in fact theWeyl group of the general linear group: the diagonal matrices are amaximal torus in the general linear group (and are their owncentralizer), the generalized permutation matrices are the normalizer of this torus, and the quotient,${\displaystyle N(T)/Z(T)=N(T)/T\cong S_n}$ is the Weyl group.

• If a nonsingular matrix and its inverse are bothnonnegative 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$${\displaystyle det(G)=det(P)\cdot det(D)=\mathrm{sgn}(\pi )\cdot d_{11}\cdot \dots \cdot d_{nn},}$$ where${\displaystyle \mathrm{sgn}(\pi )}$ is thesign of thepermutation${\displaystyle \pi }$ associated with${\displaystyle P}$ and${\displaystyle d_{11},\dots ,d_{nn}}$ are the diagonal elements of${\displaystyle D}$.

One can generalize further by allowing the entries to lie in aring, rather than in a field. In that case if the non-zero entries are required to beunits in the ring, one again obtains a group. On the other hand, if the non-zero entries are only required to be non-zero, but not necessarily invertible, this set of matrices forms asemigroup instead.

One may also schematically allow the non-zero entries to lie in a groupG, with the understanding that matrix multiplication will only involve multiplying a single pair of group elements, not "adding" group elements. This is anabuse of notation, since element of matrices being multiplied must allow multiplication and addition, but is suggestive notion for the (formally correct) abstract group${\displaystyle G\wr S_n}$ (the wreath product of the groupG by the symmetric group).

Asigned permutation matrix is a generalized permutation matrix whose nonzero entries are ±1, and are theinteger generalized permutation matrices with integer inverse.

• It is theCoxeter group${\displaystyle B_n}$, and hasorder${\displaystyle 2^{n}n!}$.
• It is the symmetry group of thehypercube and (dually) of thecross-polytope.
• Itsindex 2 subgroup of matrices with determinant equal to their underlying (unsigned) permutation is the Coxeter group${\displaystyle D_n}$ and is the symmetry group of thedemihypercube.
• It is a subgroup of theorthogonal group.

Monomial matrices occur inrepresentation theory in the context ofmonomial representations. A monomial representation of a groupG is a linear representationρ :G → GL(n,F) ofG (hereF is the defining field of the representation) such that theimageρ(G) is a subgroup of the group of monomial matrices.

• Joyner, David (2008).Adventures in group theory. Rubik's cube, Merlin's machine, and other mathematical toys (2nd updated and revised ed.). Baltimore, MD: Johns Hopkins University Press.ISBN 978-0-8018-9012-3.Zbl 1221.00013.

• Matrices (mathematics)
• Permutations
• Sparse matrices

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