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In this letter, we investigate the optimal structure of the source precoding matrix and the relay amplifying matrix for non-regenerative multiple-input multiple-output (MIMO) relay communication systems with the direct source-destination link. We show that both the optimal source precoding matrix and the optimal relay amplifying matrix have a beamforming structure. Based on this structure, an iterative joint source and relay beamforming algorithm is developed to minimize the mean-squared error (MSE) of the signal waveform estimation. Numerical example demonstrates an improved performance of the proposed algorithm.