Scalar Poincaré implies matrix Poincaré

Abstract

We prove that every reversible Markov semigroup which satisfies a Poincaré inequality satisfies a matrix-valued Poincaré inequality for Hermitian d×d matrix valued functions, with the same Poincaré constant. This generalizes recent results [ABY19, Kat20] 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.