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This paper studies the transmit power-multiplexing tradeoff in MIMO systems when two users with different priorities are to be accommodated simultaneously. In the general problem formulation, a minimum information rate must be guaranteed for a prioritized user whose channel state information (CSI) is known, while the data rate for the other user with completely unknown CSI is "best-effort." The drawbacks of the classic spatial water-filling algorithm are discussed, and two modified water-filling algorithms are proposed that either balance the required transmit power with the number of spatial dimensions available for each user, or minimize the number of transmit spatial dimensions allocated for the prioritized user. We then specialize to applications in the MIMO wiretap channel, the MIMO multiuser downlink channel, and spectrum underlay in MIMO cognitive radio networks, where existing methods are not applicable under the asymmetric CSI scenario. Numerical results verify the performance advantages of the proposed transmission schemes for each of these case studies.