On singular values of products of matrices and log-majorization

Abstract

In this paper, we are concerned with the problem of improving the classical inequality $s ( X Y ) \prec _{\log } s (X) \circ s (Y )$, where X, Y are $n \times n$ matrices, s(A) denotes the vector of singular values of a matrix A, $\circ$ is the Schur product on ${{\mathbb {R}}}^{n}$ and $\prec _{\log }$ stands for the log-majorization preorder on ${{\mathbb {R}}}^{n}$. We show that $s ( X Y ) \prec _{\log } s (X Z) \circ s (W ) \prec _{\log } s (X) \circ s (Y )$ for some special matrices Z and W depending on Y. Moreover, we prove that the operator $Z \mapsto s (X Z) \circ s (Z)^{[-1]} \circ s (Y)$ is monotone. To this end we introduce an adequate preorder on the matrix space ${{\mathbb {M}}}_{n}$. Some related results are also demonstrated.