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

Abstract EN

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 A¯=RAR, where A¯ 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.