Dual quaternions have important applications in fields such as information control, robotics, and hand-eye calibration. At the same time, matrix equations play a crucial role in system control, particularly the generalized Sylvester matrix equation AX+ EXF= CY+ D, which has extensive applications in higher-order linear systems. However, research on this matrix equation in the context of dual quaternions has not yet been discovered. Therefore, this paper aims to fill this research gap by establishing the necessary and sufficient conditions for the solvability of this generalized Sylvester matrix equation over dual quaternions and providing a general solution when it is consistent. As an application, we design a color image encryption and decryption scheme based on this generalized Sylvester matrix equation. Experimental results demonstrate the high feasibility and effectiveness of the proposed scheme.

In comparison to quaternions, split quaternions exhibit a more intricate algebraic structure, characterized by the presence of nontrivial zero factors. Furthermore, in various fields such as geometry and electromagnetism, split quaternions serve as more convenient research tools than quaternions. This paper aims to establish the necessary and sufficient conditions for the existence of (anti-) η-Hermitian solutions to a novel system of matrix equations formulated over split quaternions. We also provide the general expression of such solutions when the system is solvable, along with the least squares (anti-) η-Hermitian solution in the case of inconsistency. To illustrate the key findings of this paper, numerical examples are presented.