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A Gaussian multiple-input multiple-output broadcast channel (MIMO GBC) is considered. Throughout the paper it is assumed that 1) input signals are Gaussian and 2) perfect channel state information is available at the transmitter and at the receivers. By considering each data stream as a single user, the uplink-downlink signal-to-interference-plus-noise (SINR) duality is generalized to the MIMO case with general cross-talk matrix. The duality is subsequently applied to finding the solution for the SINR-balancing problem. The result serves as a tool for characterizing achievable rate regions of different coding strategies. Next, we investigate a superposition coding scheme proposed by Cover-van der Meulen-Hajek and Pursley (nicknamed CMHP), where there is a common message to both users. We consider a MIMO broadcast channel with two users, each user has two antennas and the transmitter has four antennas. Assuming one common stream is sent by CMHP coding and successive decoding, a lower bound to the CMHP rate region is found. Behaviors of the CMHP rate region and sumrate are analyzed. We find the sumrate gaps between DPC, CMHP, and MMSE at high SNR for general 2-user multiple-input singleoutput (MISO) Gaussian broadcast channel. The result suggests when CMHP is beneficial for sumrate.