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Abstract We give Hoeffding- and Bernstein-type concentration inequalities for the largest eigenvalue of sums of random matrices arising from a Markov chain. We consider time-dependent matrix-valued functions on a general state space, generalizing previous results that had only considered Hoeffding-type inequalities, and only for time-independent functions on a finite state space. In particular, we study a kind of non-commutative moment generating function, provide tight bounds on this object and use a method of Garg et al. (A matrix expander Chernoff bound. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, page 1102–1114, New York, NY, USA, 2018. Association for Computing Machinery) to turn this into tail bounds. Our proof proceeds spectrally, bounding the norm of a certain perturbed operator. In the process we make an interesting connection to dynamical systems and Banach space theory to prove a crucial result on the limiting behaviour of our moment generating function that may be of independent interest.