Reconfigurable intelligent surfaces (RISs) represent a new technology that can shape the radio wave propagation in wireless networks and offers a great variety of possible performance and implementation gains. Motivated by this, we study the achievable rate optimization for multi-stream multiple-input multiple-output (MIMO) systems equipped with an RIS, and formulate a joint optimization problem of the covariance matrix of the transmitted signal and the RIS elements. To solve this problem, we propose an iterative optimization algorithm that is based on the projected gradient method (PGM). We derive the step size that guarantees the convergence of the proposed algorithm and we define a backtracking line search to improve its convergence rate. Furthermore, we introduce the total free space path loss (FSPL) ratio of the indirect and direct links as a first-order measure of the applicability of RISs in the considered communication system. Simulation results show that the proposed PGM achieves the same achievable rate as a state-of-the-art benchmark scheme, but with a significantly lower computational complexity. In addition, we demonstrate that the RIS application is particularly suitable to increase the achievable rate in indoor environments, as even a small number of RIS elements can provide a substantial achievable rate gain.