In this paper, we study moment and concentration inequalities for the spectral norm of sums of dependent random matrices. We establish novel Rosenthal-Burkholder inequalities for discrete-time matrix local martingales, Burkholder-Davis-Gundy inequality for continuous matrix local martingales, as well as matrix Rosenthal, Hoeffding, and Bernstein inequalities for ergodic Markov chains. Compared with previous work on matrix concentration inequalities for Markov chains, which assume a non-zero absolute $L^2$-spectral gap or the stronger $ψ$-mixing condition, our results assume geometric ergodicity, a condition commonly used in statistical applications. Furthermore, our results have leading terms that match the Markov chain central limit theorem, rather than relying on suboptimal variance proxies. We also give dimension-free versions of the inequalities, which are independent of the ambient dimension $d$ and relies on the effective rank instead. This enables the generalization of our results to linear operators in infinite-dimensional Hilbert spaces. Our results have extensive applications in statistics and machine learning; in particular, we obtain improved bounds in covariance estimation and principal component analysis on Markovian data.