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Psychometrika

, Volume 56, Issue 4, pp 631–636 | Cite as

Kruskal's polynomial for 2×2×2 arrays and a generalization to 2×n×n arrays

  • Jos M. F. ten Berge
Article

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 ofn×n matrices that have a rank less thann 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 twon ×n matrices can be diagonalized simultaneously with positive probability.

Key words

rank three-way arrays PARAFAC CANDECOMP simultaneous diagonalization 

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References

  1. Harshman, R. A. (1972).Determination and proof of minimum uniqueness conditions for PARAFACL (UCLA Working Papers in Phonetics, No. 22, pp. 111–117). Los Angeles: UCLA.Google Scholar
  2. 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.Google Scholar
  3. Kruskal, J. B. (1983).Statement of some current results about three-way arrays. Unpublished manuscript, AT&T Bell Laboratories, Murray Hill, NJ.Google Scholar
  4. Kruskal, J. B. (1989). Rank, decomposition, and uniqueness for 3-way and N-way arrays. In R. Coppi & S. Bolasco (Eds.),Multiway data analysis (pp. 7–18). Amsterdam: North-Holland.Google Scholar

Copyright information

© The Psychometric Society 1991

Authors and Affiliations

  • Jos M. F. ten Berge
    • 1
  1. 1.University of GroningenThe Netherlands

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