The Sylvester-type equation has applications in many fields, such as control theory, scientific computing, and signal processing. However, research on the Sylvester-type equation over dual quaternions remains limited. In this paper, using generalized inverses and matrix rank, we provide necessary and sufficient conditions for the solvability of the dual quaternion matrix equation AX+BYC+ZD=E. When the solvability conditions are satisfied, the general expression of the solution is derived. Subsequently, a numerical example is presented to verify the obtained results. Finally, based on these results, an application in color image encryption and decryption is provided.
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Li et al. (2026) studied this question.
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