Iterative Methods to Solve the Generalized Coupled Sylvester-Conjugate Matrix Equations for Obtaining the Centrally Symmetric (Centrally Antisymmetric) Matrix Solutions

Abstract

The iterative method is presented for obtaining the centrally symmetric (centrally antisymmetric) matrix pair $(X,Y)$ solutions of the generalized coupled Sylvester-conjugate matrix equations $A_{\mathrm{1}}X+B_{\mathrm{1}}Y=D_{\mathrm{1}}\overline{X}{E}_{\mathrm{1}}+F_{\mathrm{1}}$, $A_{\mathrm{2}}Y+B_{\mathrm{2}}X=D_{\mathrm{2}}\overline{Y}{E}_{\mathrm{2}}+F_{\mathrm{2}}$. On the condition that the coupled matrix equations are consistent, we show that the solution pair $(X^{\mathrm{*}},Y^{\mathrm{*}})$ can be obtained within finite iterative steps in the absence of round-off error for any initial value given centrally symmetric (centrally antisymmetric) matrix. Moreover, by choosing appropriate initial value, we can get the least Frobenius norm solution for the new generalized coupled Sylvester-conjugate linear matrix equations. Finally, some numerical examples are given to illustrate that the proposed iterative method is quite efficient.