## Journal of Applied Mathematics

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

#### Abstract

Two efficient iterative algorithms are presented to solve a system of matrix equations ${A}_{1}{X}_{1}{B}_{1}$ + ${A}_{2}{X}_{2}{B}_{2}$ $=E$, ${C}_{1}{X}_{1}{D}_{1}$ + ${C}_{2}{X}_{2}{D}_{2}$ $=F$ over generalized reflexive and generalized antireflexive matrices. By the algorithms, the least norm generalized reflexive (antireflexive) solutions and the unique optimal approximation generalized reflexive (antireflexive) solutions to the system can be obtained, too. For any initial value, it is proved that the iterative solutions obtained by the proposed algorithms converge to their true values. The given numerical examples demonstrate that the iterative algorithms are efficient.

#### Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 352327, 9 pages.

Dates
First available in Project Euclid: 2 March 2015

https://projecteuclid.org/euclid.jam/1425305544

Digital Object Identifier
doi:10.1155/2014/352327

Mathematical Reviews number (MathSciNet)
MR3176816

Zentralblatt MATH identifier
07010609

#### Citation

Lin, Yong; Wang, Qing-Wen. Generalized Reflexive and Generalized Antireflexive Solutions to a System of Matrix Equations. J. Appl. Math. 2014 (2014), Article ID 352327, 9 pages. doi:10.1155/2014/352327. https://projecteuclid.org/euclid.jam/1425305544

#### 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.
• M. Dehghan and M. Hajarian, “An iterative algorithm for solving a pair of matrix equations $AY B=E$, $CY D=F$ over generalized centro-symmetric matrices,” Computers & Mathematics with Applications, vol. 56, no. 12, pp. 3246–3260, 2008.
• 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.
• A. L. Andrew, “Solution of equations involving centrosymmetric matrices,” Technometrics, vol. 15, no. 2, pp. 405–407, 1973.
• 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.
• 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.
• X. Sheng and G. Chen, “A finite iterative method for solving a pair of linear matrix equations $(AXB,CXD )=(E,F)$,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1350–1358, 2007.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• A.-P. Liao and Y. Lei, “Least-squares solution with the minimum-norm for the matrix equation $(AXB,GXH )=(C,D)$,” Computers & Mathematics with Applications, vol. 50, no. 3-4, pp. 539–549, 2005.
• 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.
• 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.
• 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.
• 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. \endinput