Iterative Hermitian $R$-Conjugate Solutions to General Coupled Sylvester Matrix Equations

Sheng-Kun Li

For a 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 $A=A^H$, $RAR=\bar A$. 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 when the considered matrix equations have a unique solution group then the proposed method is always convergent for any initial Hermitian $R$-conjugate matrix group under a loose restriction on the convergent factor. Furthermore, the optimal convergent factor is derived. Finally, two numerical examples are given to demonstrate the theoretical results and effectiveness.