Canonical Forms of 2x2x2 and 2x2x2x2 Tensors

Abstract

The rank and canonical forms of a tensor are concepts that naturally generalize that of a matrix. The question of how to determine the rank of a tensor has been widely studied in the literature and has no known solution in general. There are only a few specific cases that are known. In particular, the maximum rank of a 2x2x2 tensor is 3. This fact was first proved by Kruskal. Later, ten Berge simplified the proof by providing a more straightforward argument. We provide another proof that is more simplified. As a corollary, a new upper bound on the rank of 2x...x2 tensors (with n > 2 factors) is 3(2^{n-3}).

For 2x2x2 tensors, we consider their canonical forms over R (real numbers), C (complex numbers), and some finite fields, F_p. We consider the direct product of the general linear groups and verify that over R, these tensors are equivalent to eight canonical forms. When we consider the same problem over C there are seven canonical forms. These results were discovered independently many times in the literature. Using computer algebra for the case of finite fields, we additionally consider the action of the semidirect product of general linear groups with the symmetric group. For each canonical form, we determine the size of its orbit, and the rank of the tensors in its orbit over F_p for p = 2, 3, 5. These are original results. For larger primes, our computer did not have sufficient memory to finish the computations.

For 2x2x2x2 tensors, a finite classification of canonical forms over R and C is not possible. Instead, we use computer algebra and consider the semidirect product of general linear groups with the symmetric group and determine the canonical forms, the size of its orbit, and the rank of the tensors in its orbit over F_p for p = 2, 3. These are original results. For larger primes, the number of canonical forms becomes too large to be publishable.