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In recent years, the research on multiple-input multiple-output (MIMO) broadcast channels has attracted much interest, especially since the discovery of the broadcast channel capacity achievable through the use of dirty paper coding (DPC). In this paper, we propose a new matrix decomposition, called the block diagonal geometric mean decomposition (BD-GMD), and develop transceiver designs that combine DPC with BD- GMD for MIMO broadcast channels. We also extend the BD- GMD to the block diagonal uniform channel decomposition (BD- UCD) with which the MIMO broadcast channel capacity can be achieved. Our proposed schemes decompose each user's MIMO channel into parallel subchannels with identical SNRs/SINRs, thus equal-rate coding can be applied across the subchannels of each user. Numerical simulations show that the proposed schemes demonstrate superior performance over conventional schemes.