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Kruskal's Polynomial for 2 × 2 × 2 Arrays and a Generalization to 2 × n × n Arrays

Published online by Cambridge University Press:  01 January 2025

Jos M. F. ten Berge
Jos M. F. ten Berge*
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Jos M. F. ten Berge, Vakgroep Psychologie, Grote Kruisstraat 2/1, 9712 TS Groningen, THE NETHERLANDS.
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Abstract

A remarkable difference between the concept of rank for matrices and that for three-way arrays has to do with the occurrence of non-maximal rank. The set of n × n matrices that have a rank less than n has zero volume. Kruskal pointed out that a 2 × 2 × 2 array has rank three or less, and that the subsets of those 2 × 2 × 2 arrays for which the rank is two or three both have positive volume. These subsets can be distinguished by the roots of a certain polynomial. The present paper generalizes Kruskal's results to 2 × n × n arrays. Incidentally, it is shown that two n × n matrices can be diagonalized simultaneously with positive probability.

Keywords

rankthree-way arraysPARAFACCANDECOMPsimultaneous diagonalization
Type
Original Paper
Information
Psychometrika , Volume 56 , Issue 4 , December 1991 , pp. 631 - 636
Copyright
Copyright © 1991 The Psychometric Society

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Footnotes

The author is obliged to Joe Kruskal and Henk Kiers for commenting on an earlier draft, and to Tom Wansbeek for raising stimulating questions.

References

Harshman, R. A. (1972). Determination and proof of minimum uniqueness conditions for PARAFACL (pp. 111–117). Los Angeles: UCLA.Google Scholar
Kruskal, J. B. (1977). Three-way arrays: Rank and uniqueness of trilinear decompositions, with applications to arithmetic complexity and statistics. Linear Algebra and its Applications, 18, 95–138.CrossRefGoogle Scholar
Kruskal, J. B. (1983). Statement of some current results about three-way arrays, Murray Hill, NJ: AT&T Bell Laboratories.Google Scholar
Kruskal, J. B. (1989). Rank, decomposition, and uniqueness for 3-way and N-way arrays. In Coppi, R., Bolasco, S. (Eds.), Multiway data analysis (pp. 7–18). Amsterdam: North-Holland.Google Scholar