  Matrix Joint Diagonalization (MJD) is a powerful approach for solving the
Blind Source Separation (BSS) problem. It relies on the construction of
matrices which are diagonalized by the unknown demixing matrix. Their joint
diagonalizer serves as a correct estimate of this demixing matrix only if it is
uniquely determined. Thus, a critical question is under what conditions a joint
diagonalizer is unique. In the present work we fully answer this question about
the identifiability of MJD based BSS approaches and provide a general result on
uniqueness conditions of matrix joint diagonalization. It unifies all existing
results which exploit the concepts of non-circularity, non-stationarity,
non-whiteness, and non-Gaussianity. As a corollary, we propose a solution for
complex BSS, which can be formulated in a closed form in terms of an eigenvalue
and a singular value decomposition of two matrices.
