The tensor rank problem over the quaternions

Abstract

In this thesis, we give the maximal rank (the best possible upper bound) of quaternionic tensors in the some small cases. Decomposition of a quaternionic tensor into simple tensors in the some of these cases are discussed. We also give an example of a complex tensor that has different ranks over the complex field and the real quaternion algebra.

Moreover, we give the maximal rank and canonical forms of arbitrary quaternionic 3-tensors with 2 frontal slices. We also discuss some upper bounds for general quaternion tensors by using a block tensor approach.