Journal of Applied Mathematics

An Iterative Algorithm for the Generalized Reflexive Solution of the Matrix Equations $\mathrm{AXB}=E$, $\mathrm{CXD}=F$

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 ${X}_{1}$, 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 $AXB=E, CXD=F$ 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 $A\stackrel{̃}{X}B=\stackrel{̃}{E}, C\stackrel{̃}{X}D=\stackrel{̃}{F}$ with $\stackrel{̃}{E}=E-A{X}_{0}B, \stackrel{̃}{F}=F-C{X}_{0}D$. Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.

Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 492951, 20 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.jam/1355495209

Digital Object Identifier
doi:10.1155/2012/492951

Mathematical Reviews number (MathSciNet)
MR2948150

Zentralblatt MATH identifier
1251.65046

Citation

Chen, Deqin; Yin, Feng; Huang, Guang-Xin. An Iterative Algorithm for the Generalized Reflexive Solution of the Matrix Equations $\mathrm{AXB}=E$ , $\mathrm{CXD}=F$. J. Appl. Math. 2012 (2012), Article ID 492951, 20 pages. doi:10.1155/2012/492951. https://projecteuclid.org/euclid.jam/1355495209