The Tensor Rank Problem over the Quaternions

YG Liang; Sergio Da Silva; Yang Zhang

doi:10.1016/j.laa.2021.02.019

paper

The Tensor Rank Problem over the Quaternions

arXiv:2006.14813 · doi:10.1016/j.laa.2021.02.019

Abstract

We provide a nontrivial bound on the rank of any tensor $T$ over the quaternions $H$ in the $n_{1} \times n_{2} \times n_{3}$ cases where $2 \leq n_{i} \leq 3$ . We describe a decomposition of $T$ into $3$ simple tensors in the $2 \times 2 \times 2$ case. We also show that the upper bound is the best possible for some of the cases, and we provide various partial results involving tensor decompositions over $C$ and $H$ .

24 pages, no figures