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arXiv · 2006.09567

Scalar Poincar\'e Implies Matrix Poincar\'e

Ankit Garg·Tarun Kathuria·Nikhil Srivastava
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Abstract

We prove that every reversible Markov semigroup which satisfies a Poincar\'e inequality satisfies a matrix-valued Poincar\'e inequality for Hermitian $d\times d$ matrix valued functions, with the same Poincar\'e constant. This generalizes recent results [Aoun et al. 2019, Kathuria 2019] establishing such inequalities for specific semigroups and consequently yields new matrix concentration inequalities. The short proof follows from the spectral theory of Markov semigroup generators.

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Ankit Garg, Tarun Kathuria, Nikhil Srivastava. 2020-06-16. Scalar Poincar\'e Implies Matrix Poincar\'e. https://arxiv.org/abs/2006.09567

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