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A Simplification of a Result by Zellini on the Maximal Rank of Symmetric Three-Way Arrays

Published online by Cambridge University Press:  01 January 2025

Roberto Rocci  and
Jos M. F. ten Berge
Roberto Rocci*
Affiliation:
University of Rome “La Sapienza”
Jos M. F. ten Berge
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Roherto Rocci, Department of Statistics, University of Rome “La Sapienza”, Piazzale Aldo Moro, 5—00185 Rome, Italy.
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Abstract

Zellini (1979, Theorem 3.1) has shown how to decompose an arbitrary symmetric matrix of order n × n as a linear combination of 1/2n(n + 1) fixed rank one matrices, thus constructing an explicit tensor basis for the set of symmetric n × 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 1/2n(n + 1) fixed binary matrices of rank one. The decomposition implies that an n × n × p array consisting of p symmetric n × n slabs has maximal rank 1/2n(n + 1). Likewise, an unconstrained 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 matrices, as is the case in a version of PARAFAC where communalities are involved, the maximal number of dimensions can be further reduced to 1/2n(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.

Keywords

INDSCALCANDECOMPPARAFACthree-way ranktensor rank
Type
Original Paper
Information
Psychometrika , Volume 59 , Issue 3 , September 1994 , pp. 377 - 380
Copyright
Copyright © 1994 The Psychometric Society

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References

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