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Assuming perfect channel state information (CSI), linear precoding/decoding for multiple-input multiple-output (MIMO) systems has been considered in the literature under various performance criteria, such as minimum total mean-square error (MSE), maximum mutual information, and minimum average bit error rate (BER). It has been shown that these criteria belong to a set of reasonable Schur-concave or Schur-convex objective functions of the diagonal entries of the system mean-square error (MSE) matrix. In this paper, assuming only the knowledge of channel mean and transmit correlation at both ends, a general theoretical framework is presented to derive the optimum precoder and decoder for MIMO systems using these objective functions. It is shown that for all these objective functions the optimum transceivers share a similar structure. Compared to the case with perfect CSI, a linear filter is added to both ends to balance the suppression of channel noise and the additional noise induced from channel estimation error. Simulation results are provided.