On the singular values of a product of matrices

The purpose of this note is to give necessary and sufficient conditions for the singular values of a product of matrices to be equal to certain products of their singular values. We then analyze the case of equa]jty in a matrix inequality of Os trows ki . Th e s ingular values of an n·square co mplex matrix X are the positive square roots of the eigen· values of X*X, where X* is the conjugate tran spose of X. Denote the singular values of X by a t (X) , ... , an (X), arranged so that al (X) ~ ... ~ an(X) > 0 (all matrices are assumed to be nonsi ngular). Let A and B be n,·square complex matri ces and let A = UH, B = VK be the polar factorizations of A a nd B. In the factorization s U and V are unitary matrices and Hand K are positive·definit e hermitian matri ces. THEOREM 1: Let k be a positive integer less than n. Then