An iterative algorithm for the generalized re”exive solutions of the general coupled matrix equations

(including the generalized coupled Sylvester matrix equations as special cases) have numerous applications in control and system theory. In this paper, an iterative algorithm is constructed to solve the general coupled matrix equations and their optimal approximation problem over generalized reflexive matrix solution (X1,X2, . . . ,Xq). When the general coupled matrix equations are consistent over generalized reflexive matrices, the generalized reflexive solution can be determined automatically by the iterative algorithm within finite iterative steps in the absence of round-off errors. The least Frobenius norm generalized reflexive solution of the general coupled matrix equations can be derived when an appropriate initial matrix group is chosen. Furthermore, the unique optimal approximation generalized reflexive solution (̂X1, X̂2, . . . , X̂q) to a given matrix group (X0 1 ,X0 2 , . . . ,X0 q ) in Frobenius norm can be derived by finding the least-norm generalized reflexive solution (̃X* 1, X̃ * 2, . . . , X̃* q) of the corresponding general coupled matrix equations ∑q j=1 Aij̃XjBij = M̃i , i = 1, 2, . . . ,p, where X̃j = Xj – X0 j , M̃i =Mi – ∑q j=1 AijX 0 j Bij . A numerical example is given to illustrate the effectiveness of the proposed iterative algorithm. MSC: 15A18; 15A57; 65F15; 65F20