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In this paper we propose low complexity optimum transmit antenna selection algorithms for wireless spatial multiplexing (SM) systems. Antenna selection with less radio frequency (RF) chains could significantly reduce the system cost and power consumption. Antenna selection criteria such as maximum minimum signal to interference plus noise ratio (SINR) and MMSE require computation of matrix inverse which has high computational complexity. Motivated by matrix inverse properties, several matrix inverse update formulas are developed for low complexity optimum transmit antenna selection. Different from the conventional optimum antenna selection algorithms, the proposed algorithms update matrix inverse based on the existed matrix inverse, so that the algorithmic complexity is reduced significantly. With greatly reduced complexity, the proposed algorithms are mathematically equivalent to the conventional optimum algorithms without any performance loss.