In this paper, the beam design for signal-space interference alignment in slowly fading multiuser multiple-input multiple-output (MIMO) interference channels is considered. Based on a linear formulation for interference alignment, a predictive beam tracking algorithm is proposed using matrix perturbation theory. The proposed algorithm, based on a mixture of iteration and update, computes interference-aligning beamforming vectors at the current time by updating the previous beam vectors based on the channel difference between the two time steps during the predictively updating phase, and yields significant reduction in computational complexity compared with existing methods recalculating beams at each time step. The tracking performance of the algorithm is analyzed in terms of mean square error and sum rate loss between the predictively updating approach and the recalculating approach, and the impact of imperfect channel knowledge is also investigated under the state-space channel model. Numerical results show that the proposed algorithm has almost the same performance as non-predictive methods in sum rate. Thus, the proposed algorithm provides a very efficient way to realize interference alignment in a realistic slowly fading MIMO channel environment.