Қатарлары мен бағаналарында бір ғана нөлдік емес элементі бар матрицалар

Matrix with one nonzero entry in each row and column
In mathematics, a generalized permutation matrix (or monomial matrix) is a 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

An invertible matrix A is a generalized permutation matrix if and only if it can be written as a product of an invertible diagonal matrix D and an (implicitly invertible) permutation matrix P: i. e.,

The subgroup where all entries are 1 is exactly the permutation matrices, which is isomorphic to the symmetric group. The subgroup where all entries are ±1 is the signed permutation matrices, which is the hyperoctahedral group. The subgroup where the entries are mth roots of unity is isomorphic to a generalized symmetric group. The subgroup of diagonal matrices is abelian, normal, and a maximal abelian subgroup. The quotient group is the symmetric group, and this construction is in fact the Weyl group of the general linear group: the diagonal matrices are a maximal torus in the general linear group (and are their own centralizer), the generalized permutation matrices are the normalizer of this torus, and the quotient, is the Weyl group.

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 where is the sign of the permutation associated with and are the diagonal elements of .

One can generalize further by allowing the entries to lie in a ring, rather than in a field. In that case if the non zero entries are required to be units 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 a semigroup instead. One may also schematically allow the non zero entries to lie in a group G, with the understanding that matrix multiplication will only involve multiplying a single pair of group elements, not "adding" group elements. This is an abuse of notation, since element of matrices being multiplied must allow multiplication and addition, but is suggestive notion for the (formally correct) abstract group (the wreath product of the group G by the symmetric group).

A signed permutation matrix is a generalized permutation matrix whose nonzero entries are ±1, and are the integer generalized permutation matrices with integer inverse.

It is the Coxeter group , and has order It is the symmetry group of the hypercube and (dually) of the cross polytope. Its index 2 subgroup of matrices with determinant equal to their underlying (unsigned) permutation is the Coxeter group and is the symmetry group of the demihypercube. It is a subgroup of the orthogonal group.

Monomial matrices occur in representation theory in the context of monomial representations. A monomial representation of a group G is a linear representation ρ : G → GL(n, F) of G (here F is the defining field of the representation) such that the image ρ(G) is a subgroup of the group of monomial matrices.