Scalar Poincar\'e Implies Matrix Poincar\'e

TL;DR

This paper proves that scalar Poincaré inequalities imply matrix-valued Poincaré inequalities for reversible Markov semigroups, leading to new matrix concentration results and extending previous specific cases.

Contribution

It generalizes scalar Poincaré inequalities to matrix-valued versions for all reversible Markov semigroups satisfying the inequality.

Findings

Establishes matrix Poincaré inequalities with the same constant as scalar ones.

Derives new matrix concentration inequalities from the generalized inequalities.

Provides a spectral theory-based proof for the generalization.

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.