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 algorithm given to attain a singular value, based on a generalization of the transition formulae, has a Gauss-Seidel form. A recursive algorithm leads to the decomposition termed SVD-k. A generalization of the Eckart-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 component analysis (PCA) over k modes, termed PTA-k, which conserves most of the properties of a PCA.