A gradient-based iterative algorithm for generalized coupled Sylvester matrix equations over generalized centro-symmetric matrices

An n × n real matrix P is said to be a symmetric orthogonal matrix if P = P − 1 = P T . An n × n real matrix X is called a generalized centro-symmetric matrix with respect to P , if X = PXP . It is obvious that every n × n matrix is also a generalized centro-symmetric matrix with respect to I (identity matrix). In the present paper, we propose a gradient-based iterative algorithm to solve the generalized coupled Sylvester matrix equations

{ A 1 X 1 B 1 + C 1 X 2 D 1 = E 1 , A 2 X 1 B 2 + C 2 X 2 D 2 = E 2 ,

over the generalized centro-symmetric matrix pair ( X 1 , X 2 ) . It is proved that the iterative method is always convergent for any initial generalized centro-symmetric matrix pair ( X 1 ( 1 ) , X 2 ( 1 ) ) . Finally, a numerical example is discussed to illustrate the results.