## Journal of Applied Mathematics

### An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations

#### Abstract

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations $(AXB-CYD,EXF-GYH)=(M,N)$, which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matrices $X$ and $Y$. When the matrix equations are consistent, for any initial generalized reflexive matrix pair $[{X}_{1},{Y}_{1}]$, the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair $[\stackrel{̂}{X},\stackrel{̂}{Y}]$ to a given matrix pair $[{X}_{0},{Y}_{0}]$ in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair $[{\stackrel{̃}{X}}^{{\ast}},{\stackrel{̃}{Y}}^{{\ast}}]$ of a new corresponding generalized coupled Sylvester matrix equation pair $(A\stackrel{̃}{X}B-C\stackrel{̃}{Y}D,E\stackrel{̃}{X}F-G\stackrel{̃}{Y}H)=(\stackrel{̃}{M},\stackrel{̃}{N})$, where $\stackrel{̃}{M}=M-A{X}_{0}B+C{Y}_{0}D,\stackrel{̃}{N}=N-E{X}_{0}F+G{Y}_{0}H$. Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

#### Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 152805, 28 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.jam/1355495265

Digital Object Identifier
doi:10.1155/2012/152805

Mathematical Reviews number (MathSciNet)
MR2965698

Zentralblatt MATH identifier
1251.65063

#### Citation

Yin, Feng; Huang, Guang-Xin. An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations. J. Appl. Math. 2012 (2012), Article ID 152805, 28 pages. doi:10.1155/2012/152805. https://projecteuclid.org/euclid.jam/1355495265