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

De Huang

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arXiv · 1808.05550

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

De Huang

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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.

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De Huang. 2018-08-10. A generalized Lieb's theorem and its applications to spectrum estimates for a sum of random matrices. https://arxiv.org/abs/1808.05550

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