## 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

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