On Kruskal ’ S Theorem

In the first part of this paper, we consider 3 × 3 × 3 arrays with complex entries, and provide a complete self-contained proof of Kruskal’s theorem that the maximum rank is 5. In the second part, we provide a complete classification of the canonical forms of 3× 3× 3 arrays over F2; in particular, we obtain explicit examples of such arrays with rank 6. In 1989, Kruskal [6, page 10] stated without proof that every 3×3×3 array with real entries has rank at most 5. A few years later, Rocci [7] circulated a simplified proof of this result, based on Kruskal’s unpublished hand-written notes. The details of this argument appear never to have been published. In sections 2–4, we consider 3× 3× 3 arrays with complex entries, and provide a complete self-contained proof that the maximum rank is 5. In section 5 we consider this problem over the field F2 with two elements. A remarkable fact, first noted by von zur Gathen [9], is that in this case there exist 3 × 3 × 3 arrays of rank 6. We provide a complete classification of the canonical forms of 3× 3× 3 arrays over F2; in particular, we obtain explicit examples of such arrays with rank 6. We use without reference many basic results on multidimensional arrays which can be found in de Silva and Lim [3] and Kolda and Bader [5]. 1. Preliminaries on 3-dimensional arrays We consider a p× q × r array X with entries in an arbitrary field F of scalars: X = [xijk ], xijk ∈ F, 1 ≤ i ≤ p, 1 ≤ j ≤ q, 1 ≤ k ≤ r. By a slice of X we mean any (2-dimensional) submatrix obtained by fixing one index. Fixing i gives a horizontal slice, fixing j gives a vertical slice, and fixing k gives a frontal slice. The matrix form of X is the p × qr matrix obtained by concatenating the frontal slices X1, . . . , Xr from left to right: X = [ X1 · · · Xr ] =  x111 · · · x1q1 · · · x11r · · · x1qr .. . . . .. · · · .. . . . · · · xp11 · · · xpq1 · · · xp1r · · · xpqr  . 2010 Mathematics Subject Classification. Primary 15A69. Secondary 15-04, 15A03, 15A21, 20G20, 20G40.