Journal of Applied Mathematics

Volume 2014 (2014), Article ID 352327, 9 pages

http://dx.doi.org/10.1155/2014/352327

Research Article

## Generalized Reflexive and Generalized Antireflexive Solutions to a System of Matrix Equations

^{1}Department of Mathematics, Shanghai University, Shanghai 200444, China^{2}School of Mathematics and Statistics, Suzhou University, Suzhou 234000, China

Received 23 March 2013; Accepted 30 December 2013; Published 23 February 2014

Academic Editor: Jinyun Yuan

Copyright © 2014 Yong Lin and Qing-Wen Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- T. Meng, “Experimental design and decision support,” in
*Expert Systems: The Technology of Knowledge Management and Decision Making for the 21st Century*, C. Leondes, Ed., vol. 1, p. 119, Academic Press, New York, NY, USA, 2001. View at Google Scholar - M. Dehghan and M. Hajarian, “An iterative algorithm for solving a pair of matrix equations $AYB=E$, $CYD=F$ over generalized centro-symmetric matrices,”
*Computers & Mathematics with Applications*, vol. 56, no. 12, pp. 3246–3260, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Dehghan and M. Hajarian, “An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation,”
*Applied Mathematics and Computation*, vol. 202, no. 2, pp. 571–588, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A. L. Andrew, “Solution of equations involving centrosymmetric matrices,”
*Technometrics*, vol. 15, no. 2, pp. 405–407, 1973. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Navarra, P. L. Odell, and D. M. Young, “Representation of the general common solution to the matrix equations ${A}_{1}X{B}_{1}={C}_{1}$ and ${A}_{2}X{B}_{2}={C}_{2}$ with applications,”
*Computers & Mathematics with Applications*, vol. 41, no. 7-8, pp. 929–935, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z.-H. Peng, X.-Y. Hu, and L. Zhang, “An efficient algorithm for the least-squares reflexive solution of the matrix equation ${A}_{1}X{B}_{1}={C}_{1}$, ${A}_{2}X{B}_{2}={C}_{2}$,”
*Applied Mathematics and Computation*, vol. 181, no. 2, pp. 988–999, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. Sheng and G. Chen, “A finite iterative method for solving a pair of linear matrix equations $\left(AXB,CXD\right)=\left(E,F\right)$,”
*Applied Mathematics and Computation*, vol. 189, no. 2, pp. 1350–1358, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A.-G. Wu, G. Feng, G.-R. Duan, and W.-J. Wu, “Finite iterative solutions to a class of complex matrix equations with conjugate and transpose of the unknowns,”
*Mathematical and Computer Modelling*, vol. 52, no. 9-10, pp. 1463–1478, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A.-G. Wu, G. Feng, G.-R. Duan, and W.-J. Wu, “Iterative solutions to coupled Sylvester-conjugate matrix equations,”
*Computers & Mathematics with Applications*, vol. 60, no. 1, pp. 54–66, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A.-G. Wu, B. Li, Y. Zhang, and G.-R. Duan, “Finite iterative solutions to coupled Sylvester-conjugate matrix equations,”
*Applied Mathematical Modelling*, vol. 35, no. 3, pp. 1065–1080, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. X. Yuan, “Least squares solutions of matrix equation $AXB=E$, $CXD=F$,”
*Journal of East China Shipbuilding Institute*, vol. 18, no. 3, pp. 29–31, 2004. View at Google Scholar · View at Zentralblatt MATH - Y.-X. Peng, X.-Y. Hu, and L. Zhang, “An iterative method for symmetric solutions and optimal approximation solution of the system of matrix equations ${A}_{1}X{B}_{1}={C}_{1}$, ${A}_{2}X{B}_{2}={C}_{2}$,”
*Applied Mathematics and Computation*, vol. 183, no. 2, pp. 1127–1137, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. K. Mitra, “Common solutions to a pair of linear matrix equations ${A}_{1}X{B}_{1}={C}_{1}$and ${A}_{2}X{B}_{2}={C}_{2}$,”
*Cambridge Philosophical Society*, vol. 74, no. 2, pp. 213–216, 1973. View at Google Scholar · View at MathSciNet - N. Shinozaki and M. Sibuya, “Consistency of a pair of matrix equations with an application,”
*Keio Science and Technology Reports*, vol. 27, no. 10, pp. 141–146, 1974. View at Google Scholar · View at MathSciNet - J. W. van der Woude,
*Freeback decoupling and stabilization for linear systems with multiple exogenous variables [Ph.D. thesis]*, Technical University of Eindhoven, Eindhoven, The Netherlands, 1987. View at MathSciNet - Q.-W. Wang, “A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity,”
*Linear Algebra and its Applications*, vol. 384, no. 1–3, pp. 43–54, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - M. Dehghan and M. Hajarian, “The reflexive and anti-reflexive solutions of a linear matrix equation and systems of matrix equations,”
*The Rocky Mountain Journal of Mathematics*, vol. 40, no. 3, pp. 825–848, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Q.-W. Wang, J.-H. Sun, and S.-Z. Li, “Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra,”
*Linear Algebra and its Applications*, vol. 353, no. 1–3, pp. 169–182, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q.-W. Wang, “Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations,”
*Computers & Mathematics with Applications*, vol. 49, no. 5-6, pp. 641–650, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - F. Ding, P. X. Liu, and J. Ding, “Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle,”
*Applied Mathematics and Computation*, vol. 197, no. 1, pp. 41–50, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - L. Xie, J. Ding, and F. Ding, “Gradient based iterative solutions for general linear matrix equations,”
*Computers & Mathematics with Applications*, vol. 58, no. 7, pp. 1441–1448, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - L. Xie, Y. Liu, and H. Yang, “Gradient based and least squares based iterative algorithms for matrix equations $AXB+C{X}^{T}D=F$,”
*Applied Mathematics and Computation*, vol. 217, no. 5, pp. 2191–2199, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Ding, Y. Liu, and F. Ding, “Iterative solutions to matrix equations of the form ${A}_{i}X{B}_{i}={F}_{i}$,”
*Computers & Mathematics with Applications*, vol. 59, no. 11, pp. 3500–3507, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - B. Zhou, J. Lam, and G.-R. Duan, “Gradient-based maximal convergence rate iterative method for solving linear matrix equations,”
*International Journal of Computer Mathematics*, vol. 87, no. 3, pp. 515–527, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Z.-Y. Li, B. Zhou, Y. Wang, and G.-R. Duan, “Numerical solution to linear matrix equation by finite steps iteration,”
*IET Control Theory & Applications*, vol. 4, no. 7, pp. 1245–1253, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z.-Y. Li, Y. Wang, B. Zhou, and G.-R. Duan, “Least squares solution with the minimum-norm to general matrix equations via iteration,”
*Applied Mathematics and Computation*, vol. 215, no. 10, pp. 3547–3562, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - B. Zhou, J. Lam, and G.-R. Duan, “On Smith-type iterative algorithms for the Stein matrix equation,”
*Applied Mathematics Letters*, vol. 22, no. 7, pp. 1038–1044, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y.-B. Deng, Z.-Z. Bai, and Y.-H. Gao, “Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations,”
*Numerical Linear Algebra with Applications*, vol. 13, no. 10, pp. 801–823, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y.-T. Li and W.-J. Wu, “Symmetric and skew-antisymmetric solutions to systems of real quaternion matrix equations,”
*Computers & Mathematics with Applications*, vol. 55, no. 6, pp. 1142–1147, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - M. Dehghan and M. Hajarian, “An efficient algorithm for solving general coupled matrix equations and its application,”
*Mathematical and Computer Modelling*, vol. 51, no. 9-10, pp. 1118–1134, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - M. Dehghan and M. Hajarian, “On the reflexive and anti-reflexive solutions of the generalised coupled Sylvester matrix equations,”
*International Journal of Systems Science*, vol. 41, no. 6, pp. 607–625, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - M. Dehghan and M. Hajarian, “The general coupled matrix equations over generalized bisymmetric matrices,”
*Linear Algebra and its Applications*, vol. 432, no. 6, pp. 1531–1552, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - B. Zhou, Z.-Y. Li, G.-R. Duan, and Y. Wang, “Weighted least squares solutions to general coupled Sylvester matrix equations,”
*Journal of Computational and Applied Mathematics*, vol. 224, no. 2, pp. 759–776, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - I. Jonsson and B. Kågström, “Recursive blocked algorithms for solving triangular systems—part I: one-sided and coupled Sylvester-type matrix equations,”
*ACM Transactions on Mathematical Software*, vol. 28, no. 4, pp. 392–415, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - B. Zhou, G.-R. Duan, and Z.-Y. Li, “Gradient based iterative algorithm for solving coupled matrix equations,”
*Systems and Control Letters*, vol. 58, no. 5, pp. 327–333, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - I. Jonsson and B. Kågström, “Recursive blocked algorithms for solving triangular systems—part II: two-sided and generalized Sylvester and Lyapunov matrix equations,”
*ACM Transactions on Mathematical Software*, vol. 28, no. 4, pp. 416–435, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A.-P. Liao and Y. Lei, “Least-squares solution with the minimum-norm for the matrix equation $\left(AXB,GXH\right)=\left(C,D\right)$,”
*Computers & Mathematics with Applications*, vol. 50, no. 3-4, pp. 539–549, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Cai and G. Chen, “An iterative algorithm for the least squares bisymmetric solutions of the matrix equations ${A}_{1}X{B}_{1}={C}_{1}$, ${A}_{2}X{B}_{2}={C}_{2}$,”
*Mathematical and Computer Modelling*, vol. 50, no. 7-8, pp. 1237–1244, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - F. Yin and G.-X. Huang, “An iterative algorithm for the least squares generalized reflexive solutions of the matrix equations $AXB=E$, $CXD=F$,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 857284, 18 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. Lin and Q. W. Wang, “Iterative solution to a system of matrix equations,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 124979, 7 pages, 2013. View at Publisher · View at Google Scholar - Y.-X. Peng, X.-Y. Hu, and L. Zhang, “An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation $AXB=C$,”
*Applied Mathematics and Computation*, vol. 160, no. 3, pp. 763–777, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus