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

From Poincar\'e Inequalities to Nonlinear Matrix Concentration