Starting from an entropy-driven reinforcement learning scheme for multi-agent environments, we develop a distributed algorithm for robust spectrum management in Gaussian multiple-input, multiple-output (MIMO) uplink channels. In continuous time, our approach to optimizing the transmitters' signal distribution relies on the method of matrix exponential learning, adjusted by an entropy-driven barrier term which generates a distributed, convergent algorithm in discrete time. As opposed to traditional water-filling methods, the algorithm's convergence speed can be controlled by tuning the users' learning rate; accordingly, entropy-driven learning algorithms in MIMO systems converge arbitrarily close to the optimum signal covariance profile within a few iterations (even for large numbers of users and/or antennas per user), and this convergence remains robust even in the presence of imperfect (or delayed) measurements and asynchronous user updates.