Iterative Hermitian R-conjugate solutions to general coupled sylvester matrix equations

For given symmetric orthogonal matrix $R$, i.e., $R^{T}=R$, $R^{2}=I$, a matrix $A\in \mathbb{C}^{n\times n}$ is termed Hermitian R-conjugate matrix if $A=A^{H}$, $RAR=\overline{A}$. In this paper, an iterative method is constructed for finding the Hermitian R-conjugate solutions of general coupled Sylvester matrix equations. Convergence analysis shows that the presented method is always convergent for any initial Hermitian R-conjugate matrix group for a loose restriction on the convergent factor. Meanwhile, the optimal convergent factor is also derived. Finally, two numerical examples are given to demonstrate the theoretical results and effectiveness.