A SINGULAR-VALUE DECOMPOSITION OF A K-WAY ARRAY FOR A PRINCIPAL COMPONENT ANALYSIS OF MULTIWAY DATA, PTA-K

Employing a tensorial approach to describe a k-way array, the singular value decomposition of this type of multiarray is established. The al gorithm given to attain a singular value, based on a generalization of the transition formulae, has a Gauss-Seidel form. A recursive algorit hm leads to the decomposition termed SVD-k. A generalization of the Ec kart-Young theorem is introduced by consideration of new rank concepts : the orthogonal rank and the free orthogonal rank. The application of this generalization in data analysis is illustrated by a principal co mponent analysis (PCA) over k modes, termed PTA-k, which conserves mos t of the properties of a PCA. (C) 1998 Elsevier Science Inc.