An efficient algorithm for solving general coupled matrix equations and its application

Abstract

The general coupled matrix equations (I) { A 11 X 1 B 11 + A 12 X 2 B 12 + ⋯ + A 1 l X l B 1 l = C 1 , A 21 X 1 B 21 + A 22 X 2 B 22 + ⋯ + A 2 l X l B 2 l = C 2 , ⋮ A l 1 X 1 B l 1 + A l 2 X 2 B l 2 + ⋯ + A l l X l B l l = C l , (including the generalized coupled Sylvester matrix equations as special cases) have nice applications in various branches of control and system theory. In this paper, by extending the idea of conjugate gradient method, we propose an efficient iterative algorithm to solve the general coupled matrix equations (I). When the matrix equations (I) are consistent, for any initial matrix group, a solution group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm solution group of the general coupled matrix equations can be derived when a suitable initial matrix group is chosen. We can use the proposed algorithm to find the optimal approximation solution group to a given matrix group ( X ̂ 1 , X ̂ 2 , … , X ̂ l ) in a Frobenius norm within the solution group set of the matrix equations (I). Also several numerical examples are given to illustrate that the algorithm is effective. Furthermore, the application of the proposed algorithm for solving the system of matrix equations { D 1 X E 1 = F 1 , ⋮ D p X E p = F p , over ( R , S ) -symmetric and ( R , S ) -skew symmetric matrices is highlighted.