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We consider transmit beamforming for single-carrier frequency-division multiple access (SC-FDMA) transmission over frequency-selective multiple-input multiple-output (MIMO) channels. The beamforming filter is optimized for minimization of the sum of the mean-squared errors (MMSEs) of the transmitted data streams after equalization. Here, MIMO minimum mean-squared error linear equalization (MMSE-LE) is considered. We prove that for SC-FDMA transmission eigen-beamforming diagonalizing the overall channel together with a nonuniform power distribution is the optimum beamforming strategy in the MMSE sense. The derived optimum power allocation is similar in spirit to classical results for the optimum continuous-time transmit filter for linear modulation formats obtained by Berger/Tufts and Yang/Roy. Moreover, we present a modification of the beamformer design to mitigate an increase of the peak-to-average power ratio (PAPR) which is in general associated with beamforming. Simulation results demonstrate the excellent performance of the proposed beamforming algorithm.