This paper describes a new approach to the problem of enhancing the performance of a multiuser multiple-input-multiple-output (MIMO) system for communication from one base station to many mobile stations in both frequency-flat and frequency-selective fading channels. This problem arises in space-division multiplexing systems with multiple users where many independent signal streams can be transmitted in the same frequency and time slot through the exploitation of multiple antennas at both the base and mobile stations. Our new approach is based on maximizing a lower bound for the product of signal-to-interference plus noise ratio (SINR) of a multiuser MIMO system. This provides a closed-form (noniterative) solution for the antenna weights for all the users, under the constraint of fixed transmit power. Our solution is shown by simulation to have better performance than previously proposed iterative or noniterative solutions. In addition, our solution requires significantly reduced complexity over a gradient search-based method that directly optimizes the product SINRs while still maintaining similar performance. Our solution assumes channel state information is present at the base station or transmitter.