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.
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The author is obliged to Joe Kruskal and Henk Kiers for commenting on an earlier draft, and to Tom Wansbeek for raising stimulating questions.
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ten Berge, J.M.F. Kruskal's polynomial for 2×2×2 arrays and a generalization to 2×n×n arrays. Psychometrika 56, 631–636 (1991). https://doi.org/10.1007/BF02294495
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DOI: https://doi.org/10.1007/BF02294495