On the maximal ranks of some third-order quaternion tensors

Abstract

A complex tensor T can be considered as a quaternion tensor. Consequently, decomposing T using simple quaternion tensors, rather than simple complex tensors, can potentially result in decompositions with a smaller rank. In this paper, we first present an example demonstrating this. Furthermore, we show that the maximal rank of a 3 × 3 × 3 quaternion tensor is 5, and in doing so provide explicit decompositions into simple tensors with several cases. Finally, we provide the maximal ranks for all $n_1\times n_2\times n_3$ quaternion tensors with $2\le n_1,n_2,n_3\le 3$.