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↦(Trk[exp⁡(H+ln⁡A)])1/k, on the convex cone of all positive definite matrices. Trk[A] denotes the k-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.