Symmetric and skew-antisymmetric solutions to systems of real quaternion matrix equations

In this paper we consider symmetric and skew-antisymmetric solutions to certain matrix equations over the real quaternion algebra HH. First, a criterion for a quaternion matrix to be symmetric and skew-antisymmetric is given. Then, necessary and sufficient conditions are obtained for the matrix equation AX=CAX=C and the following system {A1X=C1XB3=C3 to have symmetric and skew-antisymmetric solutions. The expressions of such solutions of the matrix equation and the system mentioned above are also given.