A simplification of a result by Zellini on the maximal rank of symmetric three-way arrays

Zellini (1979, Theorem 3.1) has shown how to decompose an arbitrary symmetric matrix of order n x n as a linear combination of~(n + 1) fixed 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 simplified tensor basis is given, by showing that a symmetric matrix can also be decomposed in terms of~n(n + 1) fixed binary matrices of rank one. The decomposition implies that an n x n x p array consisting of p symmetric n × n slabs has maximal rank Ln(n + 1). Likewise, an unconstrained INDSCAL (symmetric CANDECOMP/PARAFAC) de~:omposition of such an array will yield a perfect fit in ln(n + 1) dimensions. When the fitting only pertains to the off-diagonal elements of the symmetric matrices, as is the case in a version of PARAFAC where communalities are involved, the maximal number of dimensions can be further reduced to ~(n - 1). However, when the saliences in INDSCAL are constrained to 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 large as p, the number of matrices analyzed.