Transforming three-way arrays to multiple orthonormality

This paper is concerned with the question to what extent the concept of row wise or columnwise orthonormality can he generalized to three-way arrays. W hereas transforming a three-way array to multiple orthogonality is immediat e, transforming it to multiple orthonormality is far from straightforward. The present paper offers an iterative algorithm for such transformations, a nd gives a proof of monotonical convergence when only two modes are orthono rmalized. Also, it is shown that a variety of three-way arrays do not permi t double orthonormalization. This is due to the order of the arrays, and ho lds regardless of the particular elements of the array. Studying three-way orthonormality has proven useful in exploring the possibilities for simplif ying the core, to guide the search for equivalent direct transformations to simplicity; see Murakami et al. (Psychometrika 1998; 63: 255-261) as an ex ample. Also, it appears in various contexts of the mathematical study of th ree-way analysis. Copyright (C) 2000 John Wiley & Sons, Ltd.