A new generalization of a system of two-sided coupled Sylvester-like quaternion tensor equations

TL;DR

This paper introduces a comprehensive framework for solving a complex system of coupled Sylvester-like quaternion tensor equations, providing conditions for solutions, specific cases, and an algorithm validated by numerical examples.

Contribution

It develops a general solution and consistency conditions for a new class of coupled quaternion tensor equations, extending previous work in tensor algebra and quaternion analysis.

Findings

Established necessary and sufficient conditions for solution existence

Derived specific cases of the coupled tensor equations

Validated the proposed algorithm with numerical examples

Abstract

This study establishes consistency conditions and a general solution for a coupled system that consists of five two-sided Sylvester-like tensor equations in ten quaternion variables throughout the Einstein tensor product. Certain specific cases are thus established. In a direct application, we investigate certain necessary and sufficient conditions for the existence of an $\eta$-Hermitian solution to five coupled two-sided Sylvester-like quaternion tensor equations. Finally, we present an algorithm and a numerical example to validate the main result.