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Abstract

In this paper, we obtain large degree strong asymptotics for matrix orthogonal polynomials with respect to exponential weights on the real line. The expansions hold for the variable $z$ in different regions of the complex plane, and the asymptotic behavior of recurrence coefficients and norms is obtained as well. The main tools are the Riemann-Hilbert formulation and the Deift-Zhou method of steepest descent, adapted to the matrix case. A central role in this analysis is played by the matrix Szeg\H{o} function, an object that has independent interest in the study of matrix orthogonal polynomials.