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MIMO systems, including those for mobile channels, are well studied as combinations of antennas, other microwave components, and sophisticated digital signal processing. The communications performance is based on many assumptions, including that of linear channels. But in practice, key components such as microwave amplifiers are non-linear. This paper addresses MIMO compensation of non-linear amplifiers at both the transmitter and the receiver. The in-band distortion is modeled using Busgang's theorem. Based on this model, a decision algorithm obtains an optimized initial estimate of the transmitted symbols which are used for subsequent channel estimation. The final channel matrix estimate is obtained through two alternative methods: statistical linearization; or analytical linearization. The approach is blind in the sense that no pilots are used for sounding the channel. Performance is evaluated by simulation, allowing comparison with the benchmark of coherent detection with perfect channel knowledge. We establish the maximum SNR value for which the nonlinearities are essentially compensated. The performances of the two methods converge as the MIMO system becomes linear. The linear case allows fair comparison with known results of the conventional decision-directed Kalman filtering, and two pilot-aided systems. Finally we compare the performance with a look-up table technique for compensating the non-linearity.