A SIMPLIFICATION OF A RESULT BY ZELLINI ON THE MAXIMAL RANK OF SYMMETRICAL 3-WAY ARRAYS

Zellini (1979, Theorem 3.1) has shown how to decompose an arbitrary sy mmetric matrix of order n x n as a linear combination of 1/2n(n + 1) f ixed rank one matrices, thus constructing an explicit tensor basis for the set of symmetric n x n matrices. Zellini's decomposition is based on properties of persymmetric matrices. In the present paper, a simpl ified tensor basis is given, by showing that a symmetric matrix can al so be decomposed in terms of 1/2n(n + 1) fixed binary matrices of rank one. The decomposition implies that an n x n x p array consisting of p symmetric n x n slabs has maximal rank 1/2n(n + 1). Likewise, an unc onstrained INDSCAL (symmetric CANDECOMP/PARAFAC) decomposition of such an array will yield a perfect fit in 1/2n(n + 1) dimensions. When the fitting only pertains to the off-diagonal elements of the symmetric m atrices, as is the case in a version of PARAFAC where communalities ar e involved, the maximal number of dimensions can be further reduced to 1/2n(n - 1). However, when the saliences in INDSCAL are constrained t o be nonnegative, the tensor basis result does not apply. In fact, it is shown that in this case the number of dimensions needed can be as l arge as p, the number of matrices analyzed.