Solving Linear Tensor Equations

TL;DR

This paper introduces a systematic method for solving linear equations involving tensors of any rank, including special cases with trace conditions, advancing the mathematical tools available for tensor analysis.

Contribution

It provides a generalized solution framework for linear tensor equations of arbitrary rank, extending previous methods to include trace conditions and specific tensor ranks.

Findings

Derived conditions for unique solutions of rank 3 tensor equations

Generalized solution method for tensors of arbitrary rank

Extended solutions to include tensor trace conditions

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.