  This paper deduces exponential matrix concentration from a Poincar\'e
inequality via a short, conceptual argument. Among other examples, this theory
applies to matrix-valued functions of a uniformly log-concave random vector.
The proof relies on the subadditivity of Poincar\'e inequalities and a chain
rule inequality for the trace of the matrix Dirichlet form. It also uses a
symmetrization technique to avoid difficulties associated with a direct
extension of the classic scalar argument.
