On the Hermitian R‐Conjugate Solution of a System of Matrix Equations
Chang-Zhou Dong, Qing‐Wen Wang, Yuping Zhang · Journal of Applied Mathematics · 2012
Let R be an n by n nontrivial real symmetric involution matrix, that is, R = R−1 = RT ≠ In. An n × n complex matrix A is termed R‐conjugate if , where 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.