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In this paper, the capacity-achieving input covariance matrices for coherent block-fading correlated multiple input multiple output (MIMO) Rician channels are determined. In contrast with the Rayleigh and uncorrelated Rician cases, no closed-form expressions for the eigenvectors of the optimum input covariance matrix are available. Classically, both the eigenvectors and eigenvalues are computed numerically and the corresponding optimization algorithms remain computationally very demanding. In the asymptotic regime where the number of transmit and receive antennas converge to infinity at the same rate, new results related to the accuracy of the approximation of the average mutual information are provided. Based on the accuracy of this approximation, an attractive optimization algorithm is proposed and analyzed. This algorithm is shown to yield an effective way to compute the capacity achieving matrix for the average mutual information and numerical simulation results show that, even for a moderate number of transmit and receive antennas, the new approach provides the same results as direct maximization approaches of the average mutual information.