The solvability conditions and exact solutions to some quaternion tensor systems

TL;DR

This paper establishes solvability conditions and explicit solutions for certain quaternion tensor systems using Moore-Penrose inverse, extending existing results in quaternion tensor algebra.

Contribution

It provides necessary and sufficient conditions for solvability and explicit general solutions for specific quaternion tensor systems, extending prior work in the field.

Findings

Derived solvability conditions for quaternion tensor systems.

Presented explicit general solutions when systems are solvable.

Extended known results in quaternion tensor algebra.

Abstract

We derive necessary and sufficient conditions for the existence of the exact solution to the Sylvester-type quaternion tensor system $ \mathcal{A}_i\ast_{N}\mathcal{X}_i+ \mathcal{Y}_i\ast_{M}\mathcal{B}_i+\mathcal{C}_i\ast_{N} \mathcal{Z}_i\ast_{M}\mathcal{D}_i+\mathcal{F}_i\ast_{N} \mathcal{Z}_{i+1}\ast_{M}\mathcal{G}_i=\mathcal{E}_i, i=\overline{1,3} $ using Moore-Penrose inverse, and present an expression of the general solution to the system when it is solvable. As an application of this system, we provide the solvability conditions and general solutions for the Sylvester-type quaternion tensor system $ \mathcal{A}_i\ast_{N}\mathcal{Z}_i\ast_{M}\mathcal{B}_i+ \mathcal{C}_i\ast_{N}\mathcal{Z}_{i+1}\ast_{M}\mathcal{D}_i= \mathcal{E}_i, i=\overline{1,4}. $ This paper can also serve as extensions to some known results.