This paper considers the joint optimization of the transmitter beamforming filters and receiver combining filters for the multiple-input multiple-output (MIMO) broadcast channel. A low-complexity iterative algorithm to compute the filters is proposed and then a closed-form expression of the filters for the downlink with two transmit antennas is derived, avoiding the need for iterative computation in this case. To enable practical implementation, a new limited feedback algorithm is proposed that exploits the channel structure in the closed form solution, and is independent of the number of receive antennas. Performance is evaluated by Monte Carlo simulations as a function of the amount of feedback. With two receive antennas, the proposed method performs close to the sum capacity of the MIMO broadcast channel, without the need for multiuser diversity and with only limited feedback.