Scalar Poincar\'e Implies Matrix Poincar\'e

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.