In this paper, we consider a reconfigurable intelligent surface (RIS)-assisted multiple-input multiple-output communication system with multiple antennas at both the base station (BS) and the user. We plan to maximize the achievable rate through jointly optimizing the transmit precoding matrix, the receive combining matrix, and the RIS reflection matrix under the constraints of the transmit power at the BS and the unit-modulus reflection at the RIS. Regarding the non-trivial problem form, we initially reformulate it into an considerable problem to make it tractable by utilizing the relationship between the achievable rate and the weighted minimum mean squared error. Next, the transmit precoding matrix, the receive combining matrix, and the RIS reflection matrix are alternately optimized. In particular, the optimal transmit precoding matrix and receive combining matrix are obtained in closed forms. Furthermore, a pair of computationally efficient methods are proposed for the RIS reflection matrix, namely the semi-definite relaxation (SDR) method and the successive closed form (SCF) method. We theoretically prove that both methods are ensured to converge, and the SCF-based algorithm is able to converges to a Karush-Kuhn-Tucker point of the problem.

14 pages, 12 figures