This paper investigates the problem of user rate balancing with power constraints for the downlink transmission of multiuser Multiple-Input-Multiple-Output (MIMO) system. However, this max-min rate problem is not jointly convex. Therefore, we transform the original problem into matrix-weighted user Mean Squared Error (MSE) balancing. In particular, we enhance user fairness by solving the min-max matrix-weighted MSE optimization problem by power allocation and choice of transceiver filters in a way to maximize the minimum downlink rate of the users. Formulating the balancing operation as constraints, we introduce the corresponding Lagrangian for which we reach the saddle point by alternating optimization. Various aspects of Perron Frobenius theory are exploited in the process. Simulation results are provided to validate the proposed algorithm and demonstrate its performance improvement over e.g. unweighted MSE balancing.