In this paper, we consider a downlink multiple-input multiple-output (MIMO) system, where a rate splitting approach is applied to serve multiple users more efficiently. We formulate a sum spectral efficiency maximization problem with respect to precoders when noisy channel state information is available at the transmitter. Finding even a suboptimal solution is a very challenging task by its non-convexity and non-smoothness. To resolve this challenge, we first approximate the non-smooth minimum function by using a LogSumExp technique, and reformulate the problem as a form of the product of Rayleigh quotients, considering channel estimation error. Then we derive the first-order optimality condition. A key observation of the obtained condition is that it can be cast as a nonlinear eigenvector-dependent eigenvalue problem so that finding a leading eigenvector is equivalent to finding a local optimal solution. To this end, we propose an algorithm inspired by a power iteration method, referred to as a generalized power iteration for rate splitting. Simulation results show that the proposed method enhances the achievable performance when channel acquisition error exists.