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Consider the max-min fairness linear transceiver design problem for a multi-user multi-input multi-output (MIMO) interference channel. When the channel knowledge is perfectly known, this problem can be formulated as the maximization of the minimum signal-to-interference-plus-noise ratio (SINR) utility, subject to individual power constraints at each transmitter. We prove in this paper that, if the number of antennas is at least two at each transmitter (receiver) and is at least three at each receiver (transmitter), the max-min fairness linear transceiver design problem is computationally intractable as the number of users becomes large. In fact, even the problem of checking the feasibility of a given set of target SINR levels is strongly NP-hard. We then propose two iterative algorithms to solve the max-min fairness linear transceiver design problem. The transceivers generated by these algorithms monotonically improve the min-rate utility and are guaranteed to converge to a stationary solution. The efficiency and performance of the proposed algorithms compare favorably with solutions obtained from the channel matched beamforming or the leakage interference minimization.