(R,S)‐Symmetric Solutions to the Generalized Quaternion Matrix Equation AXB+CYD=E

ABSTRACT This article focuses on the ‐symmetric, ‐skew symmetric solutions and least‐squares ‐(skew) symmetric solutions of the generalized quaternion matrix equation . By using the real representation method of generalized quaternion matrices and the block decomposition technique of involutory matrices, we transform the original quaternion matrix equation into an equivalent real matrix equation. The necessary and sufficient rank conditions for the existence of ‐symmetric solutions are derived, and the explicit general solution formulas are obtained. Meanwhile, the least‐squares ‐symmetric and ‐skew‐symmetric solution formulas are established by means of the Moore–Penrose inverse. A numerical example is provided to verify the correctness and effectiveness of all the proposed theoretical results. This work extends the theory of structured solutions for quaternion matrix equations.