Driven by the growing demand for higher spectral efficiency in wireless communications, intelligent reflecting surfaces (IRS) have attracted considerable attention for their ability to dynamically reconfigure the propagation environment. This work addresses the spectral efficiency maximization problem in IRS-assisted multiple-input multiple-output (MIMO) systems, which involves the joint optimization of the transmit precoding matrix and the IRS phase shift configuration. This problem is inherently challenging due to its non-convex nature. To tackle it effectively, we introduce a computationally efficient algorithm, termed ADMM-APG, which integrates the alternating direction method of multipliers (ADMM) with the accelerated projected gradient (APG) method. The proposed framework decomposes the original problem into tractable subproblems, each admitting a closed-form solution while maintaining low computational complexity. Simulation results demonstrate that the ADMM-APG algorithm consistently surpasses existing benchmark methods in terms of spectral efficiency and computational complexity, achieving significant performance gains across a range of system configurations.