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

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 x n matrices that have a rank less than n has zero volume. Kruskal pointed out that a 2 x 2 x 2 array has rank three or less, and that the subsets of those 2 x 2 x 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 x n x n arrays. Incidentally, it is shown that two n x n matrices can be diagonalized simultaneously with positive probability.