An Iterative Algorithm for the Generalized Reflexive Solution of the Matrix Equations AXB=E, CXD=F

Abstract

An iterative algorithm is constructed to solve the linear matrix equation pair AXB=E, CXD=F over generalized reflexive matrix X. When the matrix equation pair AXB=E, CXD=F is consistent over generalized reflexive matrix X, for any generalized reflexive initial iterative matrix X1, the generalized reflexive solution can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors. The unique least-norm generalized reflexive iterative solution of the matrix equation pair can be derived when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate solution of AXB=E, CXD=F for a given generalized reflexive matrix X0 can be derived by finding the least-norm generalized reflexive solution of a new corresponding matrix equation pair AX̃B=Ẽ, CX̃D=F̃ with Ẽ=E-AX0B, F̃=F-CX0D. Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.