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

Abstract

Let $R$ be an $n$ by $n$ nontrivial real symmetric involution matrix, that is,$R=R^{-1}=R^T\ne I_n$. An $n\times n$ complex matrix $A$ is termed $R$-conjugate if$\overline{A}=RAR$, where $\overline{A}$ denotes the conjugate of $A$. We give necessary and sufficientconditions for the existence of the Hermitian $R$-conjugate solution to the systemof complex matrix equations $AX=C\hspace{0.17em}\text{and}\hspace{0.17em}XB=D$ and present an expression ofthe Hermitian $R$-conjugate solution to this system when the solvability conditionsare satisfied. In addition, the solution to an optimal approximation problem isobtained. Furthermore, the least squares Hermitian $R$-conjugate solution with theleast norm to this system mentioned above is considered. The representation ofsuch solution is also derived. Finally, an algorithm and numerical examples aregiven.