Joint Eigenvalue Decomposition of Non-Defective Matrices Based on the LU Factorization With Application to ICA

Abstract

In this paper we propose a fast and efficient Jacobi-like approach named JET (Joint Eigenvalue decomposition based on Triangular matrices) for the Joint EigenValue Decomposition (JEVD) of a set of real or complex non-defective matrices based on the LU factorization of the matrix of eigenvectors. Contrarily to classical Jacobi-like JEVD methods, the iterative procedure of the JET approach can be reduced to the search for only one of the two triangular matrices involved in the factorization of the matrix of eigenvectors, hence decreasing the numerical complexity. Two variants of the JET technique, namely JET-U and JET-O, which correspond to the optimization of two different cost functions are described in detail and these are extended to the complex case. Numerical simulations show that in many practical cases the JET approach provides more accurate estimation of the matrix of eigenvectors than its competitors and that the lowest numerical complexity is consistently achieved by the JET-U algorithm. In addition, we illustrate in the ICA context the interest of being able to solve efficiently the (non-orthogonal) JEVD problem. More particularly, based on our JET-U algorithm, we propose a more robust version of an existing ICA method, named MICAR-U. The identifiability of the latter is studied and proved under some conditions. Computer results given in the context of brain interfaces show the better ability of MICAR-U to denoise simulated electrocortical data compared to classical ICA techniques for low signal to noise ratio values.