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This paper is concerned with the advantages of multi-user concurrent transmission in multiple-input multiple-output (MIMO) systems with rate constraints. We first study a maximum eigenmode beamforming (MEB) strategy for fading multiple access channels (MACs). This strategy allows each user to transmit only in its maximum eigenmode direction and applies a suboptimal matched-filter receiver with successive interference cancellation at the base station (BS). We derive a closed-form expression for the average minimum transmitted sum power required by MEB. Based on this, we show that: a) the MEB strategy is asymptotically optimal when the number of simultaneous users is sufficiently large; b) multi-user concurrent transmission has a power advantage, referred to as multi-user gain, over orthogonal transmission approaches such as time-division multiple-access; c) the number of antennas at the BS has a far stronger impact on the system performance than that at each user side. These properties are verified by simulation. Both numerical analyses and simulation results show that a major part of multi-user gain can be achieved in practical environments even with a quite small number of simultaneous users. We also study the MEB strategy for MIMO broadcast channels (BCs). The dirty paper coding (DPC) technique is necessary in this case. It is analytically shown that most observations made for MIMO MACs are extendable to MIMO BCs.