R. Rocci et Jmf. Tenberge, A SIMPLIFICATION OF A RESULT BY ZELLINI ON THE MAXIMAL RANK OF SYMMETRICAL 3-WAY ARRAYS, Psychometrika, 59(3), 1994, pp. 377-380
Citations number
8
Categorie Soggetti
Social Sciences, Mathematical Methods","Psychologym Experimental","Mathematical, Methods, Social Sciences
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.