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This paper studies the power allocation problem for a cognitive radio (CR) multiple-input-multiple-output (MIMO) system under spectrum sharing with an existing primary radio network. In particular, the ergodic capacity maximization problem is studied in the Rayleigh fading cognitive MIMO channel under both the total transmit power constraint and a set of interference power constraints where each is applied at one of the primary receivers (PRs). This paper considers that the channel from primary transmitter (PT) to secondary receiver (SR), and the channel from secondary transmitter (ST) to SR are both in short-term fading due to the mobility of SR, whereas the channels from ST to PRs are in long-term fading. By using the Lagrangian dual method, the optimal transmit precoding matrix is derived, and some nearly optimal power allocation algorithms are proposed to maximize the bounds of secondary user ergodic capacity. Performance analysis and simulation results show that the proposed schemes can approach the comparative capacity of the optimal power allocation algorithm, which has perfect instant channel state information (CSI), particularly when the antenna number of the ST exceeds the total antenna number of PRs.