On the Hermitian R-Conjugate Solution of a System of Matrix Equations

Let R be an n by n nontrivial real symmetric involution matrix, that is, R = R-1 = R-T not equal I-n. An n x n complex matrix A is termed R-conjugate if (A) over bar = RAR, where (A) over bar denotes the conjugate of A. We give necessary and sufficient conditions for the existence of the Hermitian R-conjugate solution to the system of complex matrix equations AX = C and XB = D and present an expression of the Hermitian R-conjugate solution to this system when the solvability conditions are satisfied. In addition, the solution to an optimal approximation problem is obtained. Furthermore, the least squares Hermitian R-conjugate solution with the 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.