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This paper considers a broadcast multiple-input multiple-output (MIMO) network with multiple users and simultaneous wireless information and power transfer (SWIPT). In this scenario, it is assumed that some users are able to harvest power from radio frequency (RF) signals to recharge batteries through wireless power transfer from the transmitter, while others are served simultaneously with data transmission. The criterion driving the optimization and design of the system is based on the weighted sum rate for the users being served with data. At the same time, constraints stating minimum per-user harvested powers are included in the optimization problem. This paper derives the structure of the optimal transmit covariance matrices in the case where both types of users are present simultaneously in the network, particularizing the results to the cases where either only harvesting nodes or only information users are to be served. The trade-off between the achieved weighted sum rate and the powers harvested by the user terminals is analyzed and evaluated using the rate-power (R-P) region. Finally, we propose a two-stage user grouping mechanism that decides which users should be scheduled to receive information and which users should be configured to harvest energy from the RF signals in each particular scheduling period, this being one of the main contributions of this paper. [ABSTRACT FROM AUTHOR]