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We present a novel linear precoder design based on the minimum pair-wise error probability (PEP) criterion for general space-time block coded (STBC) MIMO systems. The channel state information at the transmitter consists of the channel mean and transmit antenna correlation matrices. Approximate joint diagonalization reveals the "average eigenstructure" shared by the mean and correlation matrices. These average eigenvectors turn out to be the left singular vectors of the precoder matrix. The right singular vectors of the precoder matrix depend on the space-time code. The singular values correspond to the power allocated to each mode, which depends on the channel statistics and the space-time code. Precoder design examples for a non-orthogonal STBCs and simulation results are provided to illustrate the achieved performance gain.