IEEE SIGNAL PROCESSING MAGAZINE [34] MAY 2014 1053-5888/14/$31.00©2014IEEE Digital Object Identifier 10.1109/MSP.2014.2298045 Date of publication: 7 April 2014 M atrix decompositions such as the eigenvalue decomposition (EVD) or the singular value decomposition (SVD) have a long history in signal processing. They have been used in spectral analysis, signal/noise subspace estimation, prin- cipal component analysis (PCA), dimensionality reduction, and whitening in independent component analysis (ICA). Very often, the matrix under consideration is the covari- ance matrix of some obser- vation signals. However, many other kinds of matrices can be encountered in signal processing problems, such as time-lagged covariance matrices, quadratic spatial time-frequency matrices [21], and matrices of higher-order statistics. In concert with this diversity, the joint diagonalization (JD) or approximate JD (AJD) of a set of matrices has been recently recognized to be instrumental in signal processing, mainly because of its importance in practical signal processing problems such as source separation, blind beamforming, image denoising, blind channel identification for multiple-input, multiple-output (MIMO) telecommunication system, Doppler-shifted echo extraction in radar, and ICA. Perhaps one of the first such algo- rithms is the joint approximate diagonalization of eigenmatrices (JADE) algorithm proposed in [8]. In this algorithm, the matri- ces under consideration are Hermitian and the considered joint diagonalizer is a unitary matrix. More recently, generalizations and/or new decompositions were found to be of considerable interest. They concern new sets of matrices, a nonuni- tary joint diagonalizer, and new decompositions. INTRODUCTION In the context of noncircular complex-valued signals, com- plex symmetric (non-Hermitian) matrices provide information that can be useful and even sufficient for blind beamforming or source separation. One exam- ple is the complementary covariance matrix, also called the pseudocovariance matrix. With such complex symmetric matrices, one ends up with jointly diagonalizing a set of matrices via either the transpose congruence transform or Hermitian congruence transform. For the special two-matrix case with one Hermitian and one complex symmetric matrix, there are particularly fast JD algo- rithms based on EVD and SVD. This article provides a comprehensive survey of matrix joint decomposition techniques in the context of source separation. More precisely, we first intend to elaborate upon the signal models leading to different useful sets of matrices and their [ Gilles Chabriel, Martin Kleinsteuber, Eric Moreau, Hao Shen, Petr Tichavsky, and Arie Yeredor ] Joint Matrices Decompositions and Blind Source Separation [ A survey of methods, identification, and applications ] IMAGE LICENSED BY INGRAM PUBLISHING