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

2018/08/10 by Huang, De
#15A15 #15A16 #15A42 #15A75 #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1808.05550

Abstract

In this paper we prove the concavity of the k-trace functions, A↦ (Tr_k[exp(H+ln A)])^1/k, on the convex cone of all positive definite matrices. Tr_k[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.

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