A finite iterative algorithm for solving the generalized (P,Q)(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 {∑i=1NAi(1)XiBi(1)=C(1),⋮∑i=1NAi(M)XiBi(M)=C(M). over the generalized (P,Q)(P,Q)-reflexive matrix Xl∈Rn×mXl∈Rn×m (Al(i)∈Rp×n,Bl(i)∈Rm×q,C(i)∈Rp×q,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)(P,Q)-reflexive matrix Xl(l=1,…,N), for any generalized (P,Q)(P,Q)-reflexive initial iterative matrices Xl(0)(l=1,…,N), the generalized (P,Q)(P,Q)-reflexive solution can be obtained within finite iterative steps in the absence of roundoff errors. The unique least-norm generalized (P,Q)(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 Yl(l=1,…,N) can be derived by finding the least-norm generalized (P,Q)(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.