A generalized Lieb's theorem and its applications to spectrum estimates for a sum of random matrices

Abstract

In this paper we prove the concavity of the $k$-trace functions, $A\mapsto ({\text{Tr}}_k[\exp (H+\ln A)]{)}^{1/k}$, on the convex cone of all positive definite matrices. ${\text{Tr}}_k[A]$ denotes the $k_{\mathrm{th}}$ elementary symmetric polynomial of the eigenvalues of $A$. As an application, we use the concavity of these $k$-trace functions to derive tail bounds and expectation estimates on the sum of the $k$ largest (or smallest) eigenvalues of a sum of random matrices.

22 pages