A Pair of Hermitian and Anti-Hermitian Solutions of a Generalized Sylvester Matrix Equation over Commutative Quaternion Algebra

In this paper, we investigate a pair of Hermitian and anti-Hermitian solutions of a generalized Sylvester matrix equation AXB+CX¯D+EYF=G over commutative quaternion algebra by using a complex representation of commutative quaternion matrices, the Kronecker product, vec-operation, and Moore–Penrose-generalized inverse. We establish the necessary and sufficient conditions for the existence of solutions. Moreover, we derive explicit expressions when they are solvable. We also provide two numerical examples to illustrate the main results.