Solving Linear Tensor Equations

Damianos Iosifidis

doi:10.3390/universe7100383

Abstract

We develop a systematic way to solve linear equations involving tensors of arbitrary rank. We start off with the case of a rank 3 tensor, which appears in many applications, and after finding the condition for a unique solution we derive this solution. Subsequently, we generalize our result to tensors of arbitrary rank. Finally, we consider a generalized version of the former case of rank 3 tensors and extend the result when the tensor traces are also included.

Highlights

In many applications a tensorial equation of the formCitation: Iosifidis, D
In order to solve this equation, one could go about and split Nαμν into its irreducible decomposition and take contractions, symmetrizations and so forth, in order to find the various pieces in terms of Bαμν and its contractions. Even though this may work in some cases, it will be a difficult task in general. This procedure will fall short quickly if one wishes to generalize the above considerations and ask for the general solution (N in terms of B) of the rank-n tensorial equation: Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
The systematic way to solve Equation (1) for N is proved. We extend this result to tensors N of arbitrary rank and solve equations of the form (2)

Summary

Introduction

In order to solve this equation, one could go about and split Nαμν into its irreducible decomposition and take contractions, symmetrizations and so forth, in order to find the various pieces in terms of Bαμν and its contractions. Even though this may work in some cases, it will be a difficult task in general. This procedure will fall short quickly if one wishes to generalize the above considerations and ask for the general solution (N in terms of B) of the rank-n tensorial equation: Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The Theorems

Conclusions