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This paper proposes a power control strategy for the uplink of cellular MIMO spatial multiplexing systems, with a linear MMSE receiver applied at the base station and a single active user per time instant. A fixed per-stream SINR target is employed that allows guaranteed QoS for delay-sensitive applications. A straightforward application of single antenna power control is not possible in the MIMO context due to coordination between receive antennas and nonlinear dependence between interference and eigenspaces of the channel matrices. Two schemes are proposed to solve the problem. The first equally allocates power to all transmit antennas. Deriving an SINR lower bound based on an eigenvalue approximation of the composite interference, allows application of the conventional single antenna power control framework to solve this problem. To improve the feasibility performance, a second scheme is proposed that adaptively allocates power on the transmit antennas, where an iterative algorithm based on game theory is used to sequentially update each user's power distribution. The optimal solution with full channel knowledge, and a practical near-optimal solution requiring only partial channel knowledge, are both derived. Numerical results show that power control, compared to supposedly optimal waterfilling strategies, actually achieves higher throughput at the low SINRs typical in cellular systems, with significantly lower overhead and complexity. Due to its better exploitation of spatial diversity and reduced transmit power (and hence reduced interference), adaptive power allocation increases the achievable SINR by an order of magnitude over equal power allocation, resulting in far better coverage.