(R,S)(R,S)-conjugate solution to a pair of linear matrix equations

Let R and S be m × m and n × n nontrivial real symmetric involutions. An m × n complex matrix A is termed (R, S  )-conjugate if A¯=RAS, where A¯ denotes the conjugate of A. In this paper, necessary and sufficient conditions are established for the existence of the (R, S)-conjugate solution to the system of matrix equations AX = C and XB = D. The expression is also presented for such solution to this system. In addition, the explicit expression of this solution to the corresponding optimal approximation problem is obtained. Furthermore, the least squares (R, S)-conjugate solution with least norm to this system mentioned above is considered. The representation of such solution is also derived. Finally an algorithm and numerical examples are given.