  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.
