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This paper derives exponential tail bounds and polynomial moment inequalitiesfor the spectral norm deviation of a random matrix from its mean value. Theargument depends on a matrix extension of Stein's method of exchangeable pairsfor concentration of measure, as introduced by Chatterjee. Recent work ofMackey et al. uses these techniques to analyze random matrices with additivestructure, while the enhancements in this paper cover a wider class ofmatrix-valued random elements. In particular, these ideas lead to a boundeddifferences inequality that applies to random matrices constructed from weaklydependent random variables. The proofs require novel trace inequalities thatmay be of independent interest.

Deriving Matrix Concentration Inequalities from Kernel Couplings