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

Huang, De (2019) A generalized Lieb's theorem and its applications to spectrum estimates for a sum of random matrices. Linear Algebra and its Applications, 579 . pp. 419-448. ISSN 0024-3795. http://resolver.caltech.edu/CaltechAUTHORS:20190617-153352760

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Abstract

In this paper we prove the concavity of the k-trace functions, A↦(Tr_k[exp(H+lnA)])^(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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/j.laa.2019.06.013DOIArticle
https://arxiv.org/abs/1808.05550arXivDiscussion Paper
Additional Information:© 2019 Published by Elsevier. Received 30 November 2018, Accepted 13 June 2019, Available online 17 June 2019. The research was in part supported by the NSF Grant DMS-1613861. The author would like to thank Joel A. Tropp for providing deep insights and rich materials in theories of random matrices and multilinear algebra. The author also gratefully appreciates the inspiring discussions with Thomas Y. Hou, Florian Schaefer, Shumao Zhang and Ka Chun Lam during the development of this paper. The kind hospitality of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI), where the early ideas of this work started, is gratefully acknowledged.
Funders:
Funding AgencyGrant Number
NSFDMS-1613861
Subject Keywords:Trace inequality; Mixed discriminants; Concavity of matrix functions; Exterior algebra; Random matrices; Spectrum estimates
Classification Code:MSC 15A75; 15A15; 15A16; 15A42
Record Number:CaltechAUTHORS:20190617-153352760
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20190617-153352760
Official Citation:De Huang, A generalized Lieb's theorem and its applications to spectrum estimates for a sum of random matrices, Linear Algebra and its Applications, Volume 579, 2019, Pages 419-448, ISSN 0024-3795, https://doi.org/10.1016/j.laa.2019.06.013. (http://www.sciencedirect.com/science/article/pii/S0024379519302654)
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:96487
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:17 Jun 2019 23:00
Last Modified:24 Jun 2019 21:39

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