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In this paper, we propose a new joint optimization of linear transmit beamforming and receive combining vectors for the multiple-input multiple-output (MIMO) broadcast channel. We consider the transmission of a single information stream to two users with two or more receive antennas. Unlike past work in which iterative computation is required to design the beamformers, we derive specific formulations for the transmit beamformers for two active users via a power iteration and a generalized eigen analysis. To enable practical implementation, a new limited feedback algorithm is proposed that exploits the structure of the algorithm to avoid full channel quantization. The feedback overhead of the proposed algorithm is independent of the number of receive antennas. Monte Carlo simulations are used to evaluate the bit error rate and the sum rate performances of the proposed algorithm. Simulation results show that the proposed method performs close to the sum capacity of the MIMO broadcast channel even with limited feedback.