Solving two-sided Sylvester quaternionic matrix equations: Theoretical insights, computational implementation, and practical applications

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.