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

Abstract In this paper, we proposed an algorithm for solving the linear systems of matrix equations { ∑ i = 1 N A i ( 1 ) X i B i ( 1 ) = C ( 1 ) , ⋮ ∑ i = 1 N A i ( M ) X i B i ( M ) = C ( M ) . over the generalized ( P , Q ) -reflexive matrix X l ∈ R n × m ( A l ( i ) ∈ R p × n , B l ( i ) ∈ R m × q , C ( i ) ∈ R p × 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 ) -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.