In this paper, the performance of an underlay multiple-input multiple-output (MIMO) cognitive radio system is analytically studied. In particular, the secondary transmitter operates in a spatial multiplexing transmission mode, while a zero-forcing (ZF) detector is employed at the secondary receiver. Additionally, the secondary system is interfered by multiple randomly distributed single-antenna primary users (PUs). To enhance the performance of secondary transmission, optimal power allocation is performed at the secondary transmitter with a constraint on the interference temperature (IT) specified by the PUs. The outage probability of the secondary receiver is explicitly derived in an exact closed-form expression. Also, some special cases of practical interest, including co-located PUs and massive MIMO, are discussed. Further, to mitigate instantaneous excessive interference onto PUs caused by the time-average IT, an iterative antenna reduction algorithm is developed for the secondary transmitter and, accordingly, the average number of transmit antennas is analytically computed. Extensive numerical and simulation results corroborate the effectiveness of our analysis.