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This paper introduces a tensor framework to solve the problem of nonunitary joint block diagonalization (JBD) of a set of real or complex valued matrices. We show that JBD can be seen as a particular case of the block-component-decomposition (BCD) of a third-order tensor. The resulting tensor model fitting problem does not require the block-diagonalizer to be a square matrix: the over- and underdetermined cases can be handled. To compute the tensor decomposition, we build an efficient nonlinear conjugate gradient (NCG) algorithm. In the over- and exactly determined cases, we show that exact JBD can be computed by a closed-form solution based on eigenvalue analysis. In approximate JBD problems, this solution can be used to efficiently initialize any iterative JBD algorithm such as NCG. Finally, we illustrate the performance of our technique in the context of independent subspace analysis (ISA) based on second-order statistics (SOS). [ABSTRACT FROM PUBLISHER]