Algebraic conditions and general solution to a system of quaternion tensor equations with applications

TL;DR

This paper derives algebraic conditions and explicit formulas for solving Sylvester-type quaternion tensor equations, including applications to systems with Hermitian unknowns, and proposes an algorithm with a numerical example.

Contribution

It provides the first comprehensive algebraic framework and explicit solutions for quaternion tensor Sylvester equations, extending previous work to more complex tensor systems.

Findings

Established necessary and sufficient algebraic conditions for solvability.

Derived explicit formulas for the general solutions using Moore-Penrose inverses.

Proposed an algorithm with a numerical example for computing solutions.

Abstract

This paper investigates the necessary and sufficient algebraic conditions to a constrained system of Sylvester-type quaternion tensor equations. An explicit formula of the general solution regarding the Moore-Penrose inverses of some block given tensors is obtained. As an application of a particular case, we establish the solvability conditions and the general solution to a system of Sylvester-type quaternion tensor equations involving $\eta$-Hermitian unknowns. An algorithm with a numerical example is proposed to compute the general solution of the main system.