On Kruskal’s theorem that every 3 × 3 × 3 array has rank at most 5

In the first part, we consider 3 × 3 × 3 arrays with real or complex entries, and provide a 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 the field F 2 with two elements; in particular, we obtain explicit examples of such arrays with rank 6.