The reflexive and Hermitian reflexive solutions of the generalized Sylvester-conjugate matrix equation

Abstract

The main purpose of this correspondence is to establish two gradient based iterative (GI) methods extending the Jacobi and Gauss-Seidel iterations for solving the generalized Sylvester-conjugate matrix equation \begin{equation*} A_1XB_1+A_2\overline{X}B_2+C_1YD_1+C_2\overline{Y}D_2=E, \end{equation*} over reflexive and Hermitian reflexive matrices. It is shown that the iterative methods, respectively, converge to the reflexive and Hermitian reflexive solutions for any initial reflexive and Hermitian reflexive matrices. We report numerical tests to show the effectiveness of the proposed approaches.