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 . Generalizations 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). Signed permutation 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. Properties 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. Applications Monomial representations Monomial matrices occur in representation theory in the context of monomial representations. A monomial representation of a group G is a linear representation 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. References Matrices Permutations Sparse matrices