The ΛS-Householder matrices

Let A,S∈Mn(C) be given. Suppose that S is nonsingular and Hermitian. Then A is ΛS-orthogonal if A∗SA=S. Let u∈Cn be such that u∗Su≠0. The ΛS-Householder matrix of u is Su≡I-tuu∗S, where . We show that det (Su)=-1, so that products of ΛS-Householder matrices have determinant ±1. Let n⩾2 and let k be positive integers with k⩽n. Set Lk≡Ik⊕-In-k. We show that every ΛLk-orthogonal matrix having determinant ±1 can be written as a product of at most 2n+2 ΛLk-Householder matrices. We also determine the possible Jordan Canonical Forms of products of two ΛLk-Householder matrices.