On Kruskal's theorem that every 3 x 3 x 3 array has rank at most 5

TL;DR

This paper proves Kruskal's theorem that 3 x 3 x 3 arrays over complex numbers have a maximum rank of 5 and classifies their forms over F_2, providing explicit examples with rank 6.

Contribution

It offers a complete proof of Kruskal's theorem for complex arrays and classifies 3 x 3 x 3 arrays over F_2, including explicit rank 6 examples.

Findings

Maximum rank of 3 x 3 x 3 arrays over complex numbers is 5

Complete classification of arrays over F_2

Explicit examples of arrays with rank 6

Abstract

In the first part of this paper, we consider 3 x 3 x 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 x 3 x 3 arrays over F_2; in particular, we obtain explicit examples of such arrays with rank 6.