A finite iterative algorithm for solving the generalized (P, Q)-reflexive solution of the linear systems of matrix equations

In this paper, we proposed an algorithm for solving the linear systems of matrix equations {Sigma(N)(i=1) A(i)((1))X(i)B(i)((1)) = C-(1), over the generalized (P, Q)-reflexive matrix X-l is an element of R-nxm (A(l)((i)) is an element of Sigma(N)(i=1) A(i)((1))X(i)B(i)((1)) = C-(M). R-pxn, B-l((i)) is an element of R-mxq, C-(i) is an element of R-pxq, l = 1, 2,..., N, i = 1, 2,..., M). According to the algorithm, the solvability of the problem can be determined automatically. When the problem is consistent over the generalized (P, Q)-reflexive matrix X-l (l = 1,..., N), for any generalized (P, Q)-reflexive initial iterative matrices X-l(0) (l = 1,..., N), the generalized (P, Q)-reflexive solution can be obtained within finite iterative steps in the absence of roundoff errors. The unique least-norm generalized (P, Q)-reflexive solution can also be derived when the appropriate initial iterative matrices are chosen. A sufficient and necessary condition for which the linear systems of matrix equations is inconsistent is given. Furthermore, the optimal approximate solution for a group of given matrices Y-l (l = 1,..., N) can be derived by finding the least-norm generalized (P, Q)-reflexive solution of a new corresponding linear system of matrix equations. Finally, we present a numerical example to verify the theoretical results of this paper. (C) 2011 Elsevier Ltd. All rights reserved.