Gradient-based iterative algorithm for solving the generalized coupled Sylvester-transpose and conjugate matrix equations over reflexive (anti-reflexive) matrices

摘要

Linear matrix equations play an important role in many areas, such as control theory, system theory, stability theory and some other fields of pure and applied mathematics. In the present paper, we consider the generalized coupled Sylvester-transpose and conjugate matrix equations T_v(X) = F_v, v = 1,2, ...,N, where X = (X_1, X_2, ...,X_p) is a group of unknown matrices and for v = 1,2, ...,N, T_v(X) = ∑_(i = 1)~p ∑_(μ = 1)~(s_1) A_(viμ)X_iB_(viμ) + ∑_(μ = 1)~(s_2) C_(viμ)X_i~TD_(viμ) + ∑_(μ - 1)~(s_3) M_(viμ)X_iN_(viμ) + ∑_(μ = 1)~(s_4) H_(viμ)X_i~HG_(viμ), in which A_(viμ), B_(viμ), C_(viμ), D_(viμ), M_(viμ), N_(viμ), H_(viμ), G_(viμ) and F_v are given matrices with suitable dimensions defined over complex number field. By using the hierarchical identification principle, an iterative algorithm is proposed for solving the above coupled linear matrix equations over the group of reflexive (anti-reflexive) matrices. Meanwhile, sufficient conditions are established which guarantee the convergence of the presented algorithm. Finally, some numerical examples are given to demonstrate the validity of our theoretical results and the efficiency of the algorithm for solving the mentioned coupled linear matrix equations.