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

摘要

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_1X_1B_1 + C_1X_2D_1 = E_1, A_2X_1B_2 + C_2X_2D_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 (I), X_2(I)). Finally, a numerical example is discussed to illustrate the results.