This article explores Sylvester quaternion matrix equations and potential applications, which are important in fields such as control theory, graphics, sensitivity analysis, and three-dimensional 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.

In this article, we investigate a class of quaternionic functional equations of the two-sided Sylvester type, which arise in areas such as control theory, robotics, signal processing, and image analysis. Although quaternionic matrix equations have been extensively studied, two-sided Sylvester systems remain particularly challenging due to the noncommutativity of quaternion multiplication and the increased structural complexity they entail. We derive general solutions to these systems by establishing the necessary and sufficient conditions for solvability, unifying and by extending previous theoretical results. We propose an efficient algorithm to compute the general solution in both full-rank and rank-deficient cases, using Moore-Penrose inverses and projection operators The practical interest of our method is demonstrated through applications in perturbation theory, image processing, and robust control. We present numerical examples to validate the proposed approach, including a case study involving a multi-joint robotic manipulator. These results highlight both the theoretical relevance and computational advantages of the proposed method.

In applied and computational mathematics, quaternions are fundamental in representing three-dimensional rotations. However, specific types of quaternionic linear matrix equations remain few explored. This study introduces new quaternionic linear matrix equations and their necessary and sufficient conditions for solvability. We employ a methodology involving lemmas and ranks of coefficient matrices to develop a novel algorithm. This algorithm is validated through numerical examples, showing its applications in advanced fields. In control theory, these equations are used for analyzing control systems, particularly for spacecraft attitude control in aerospace engineering and for control of arms in robotics. In quantum computing, quaternionic equations model quantum gates and transformations, which are important for algorithms and error correction, contributing to the development of fault-tolerant quantum computers. In signal processing, these equations enhance multidimensional signal filtering and noise reduction, with applications in color image processing and radar signal analysis. We extend our study to include cases of $\eta$-Hermitian and i-Hermitian solutions. Our work represents an advancement in applied mathematics, providing computational methods for solving quaternionic matrix equations and expanding their practical applications.