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This paper presents a novel adaptive reduced-rank multiple-input-multiple-output (MIMO) equalization scheme and algorithms based on alternating optimization design techniques for MIMO spatial multiplexing systems. The proposed reduced-rank equalization structure consists of a joint iterative optimization of the following two equalization stages: 1) a transformation matrix that performs dimensionality reduction and 2) a reduced-rank estimator that retrieves the desired transmitted symbol. The proposed reduced-rank architecture is incorporated into an equalization structure that allows both decision feedback and linear schemes to mitigate the interantenna (IAI) and intersymbol interference (ISI). We develop alternating least squares (LS) expressions for the design of the transformation matrix and the reduced-rank estimator along with computationally efficient alternating recursive least squares (RLS) adaptive estimation algorithms. We then present an algorithm that automatically adjusts the model order of the proposed scheme. An analysis of the LS algorithms is carried out along with sufficient conditions for convergence and a proof of convergence of the proposed algorithms to the reduced-rank Wiener filter. Simulations show that the proposed equalization algorithms outperform the existing reduced- and full- algorithms while requiring a comparable computational cost.