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 $AXB=E,\hspace{0.17em}CXD=F$ over generalized reflexive matrix $X$. When the matrix equation pair $AXB=E,\hspace{0.17em}CXD=F$ is consistent over generalized reflexive matrix $X$, for any generalized reflexive initial iterative matrix $X_1$, 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 $AXB=E,\hspace{0.17em}CXD=F$ for a given generalized reflexive matrix $X_0$ can be derived by finding the least-norm generalized reflexive solution of a new corresponding matrix equation pair $A\tilde{X}B=\tilde{E},\hspace{0.17em}C\tilde{X}D=\tilde{F}$ with $\tilde{E}=E-A{X}_{0}B,\hspace{0.17em}\tilde{F}=F-C{X}_{0}D$. Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.