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This letter addresses the problem of the nonunitary joint block diagonalization of a given set of complex matrices whose potential applications stem from the blind separation of convolutive mixtures of sources and from the array processing. The proposed algorithm is based on the algebraic optimization of a least-mean-square criterion. One of its advantage is that a pre-whitening stage is no more compulsorily required when this algorithm is applied in the blind source separation context. Computer simulations are provided in order to illustrate its behavior in three cases: when exact block-diagonal matrices are built, then when they are progressively perturbed by an additive Gaussian noise and, finally, in the context of blind separation of convolutive mixtures of temporally correlated sources with estimated correlation matrices. A comparison with a classical orthogonal joint block diagonalization algorithm is also performed, and a new performance index is introduced to measure the performance of the separation.