About the maximal rank of 3-tensors over the real and the complex number field

TL;DR

This paper investigates the maximal rank of 3-tensors over real and complex fields, extending previous complex results to real cases with new bounds and theoretical insights.

Contribution

It provides new bounds for the maximal rank of 3-tensors over the real field, building on and extending the Atkinson-Stephens and Atkinson-Lloyd methods.

Findings

Derived bounds for real 3-tensor ranks

Extended complex tensor rank results to real tensors

Proved lemmas and propositions for real tensor analysis

Abstract

High dimensional array data, tensor data, is becoming important in recent days. Then maximal rank of tensors is important in theory and applications. In this paper we consider the maximal rank of 3 tensors. It can be attacked from various viewpoints, however, we trace the method of Atkinson-Stephens(1979) and Atkinson-Lloyd(1980). They treated the problem in the complex field, and we will present various bounds over the real field by proving several lemmas and propositions, which is real counterparts of their results.