Solvability and algorithm for Sylvester-type quaternion matrix equations with potential applications

This article explores Sylvester quaternion matrix equations and potential applications, which are important in fields such as control theory, graphics, sensitivity analysis, and threedimensional rotations. Recognizing that the determination of solutions and computational methods for these equations is evolving, our study contributes to the area by establishing solvability conditions and providing explicit solution formulations using generalized inverses. We also introduce an algorithm that utilizes representations of quaternion Moore-Penrose inverses to improve computational efficiency. This algorithm is validated with a numerical example, demonstrating its practical utility. Additionally, our findings offer a generalized framework in which various existing results in the area can be viewed as specific instances, showing the breadth and applicability of our approach. Acknowledging the challenges in handling large systems, we propose future research focused on further improving algorithmic efficiency and expanding the applications to diverse algebraic structures. Overall, our research establishes the theoretical foundations necessary for solving Sylvester-type quaternion matrix equations and introduces a novel algorithmic solution to address their computational challenges, enhancing both the theoretical understanding and practical implementation of these complex equations.