In this paper, we study multi-user multi-antenna wireless power transfer (WPT) systems, where each antenna at the energy harvesting (EH) nodes is connected to a dedicated non-linear rectifier. We propose an optimal transmit strategy which maximizes a weighted sum of the average harvested powers at the EH nodes under a constraint on the power budget of the transmitter. First, for multiple-input single-output (MISO) WPT systems, we show that it is optimal to transmit scalar symbols with an arbitrary phase and an amplitude, whose probability density function (pdf) has at most two mass points, using maximum ratio transmission (MRT) beamforming. Then, we prove that for single-input multiple-output (SIMO) WPT systems, the optimal transmit symbol amplitudes are discrete random variables, whose pdf also has no more than two mass points. For general multi-user MIMO WPT systems, we show that the optimal transmit strategy involves scalar unit-norm symbols with arbitrary phase and at most two beamforming vectors. In order to determine these vectors, we formulate a non-convex optimization problem and obtain an optimal solution based on monotonic optimization. Since the computational complexity of the optimal solution is high, we also propose a low-complexity iterative algorithm to obtain a suboptimal solution, which achieves nearly optimal performance. Our simulation results reveal that the proposed transmit strategy for multi-user MIMO WPT systems outperforms two baseline schemes, which are based on a linear EH model and a single beamforming vector, respectively. For a given transmit power budget, we show that the harvested power saturates when increasing the number of transmit antennas. Finally, we observe that the harvested power region spanned by multiple EH nodes is convex and the power harvested at one EH node can be traded for a higher harvested power at the other nodes.