  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.
