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The anti-reflexive solutions for the matrix equation \( AV + BW = EVF + C \)

  • Mohamed A. Ramadan
  • Naglaa M. El-ShazlyEmail author
  • Basem I. Selim
Article
  • 52 Downloads

Abstract

In this paper the generalized anti-reflexive solutions for the matrix equation \( AV + BW = EVF + C \) are presented. A new finite iterative algorithm is proposed for obtaining anti-reflexive solutions of this matrix equation. By this iterative method, the solvability of the anti-reflexive solutions for the matrix equation can be determined automatically. Additionally, the nearest solution to a given matrix in Frobenius norm is considered. Furthermore, the related optimal approximation problem to a given generalized Sylvester matrix equation can be derived. Finally, numerical examples are given to illustrate the effectiveness of the proposed algorithm.

Keywords

The generalized Sylvester equation Finite iterative method Anti-reflexive matrices The least Frobenius norm Optimal approximation solution 

Mathematics Subject Classification

65F10 65F30 15A06 

Notes

Acknowledgements

The authors are grateful to the referees for their valuable suggestions and comments for the improvement of the paper.

References

  1. Chen HC (1998) Generalized reflexive matrices: special properties and applications. SIAM J Matrix Anal Appl 19:140–153MathSciNetCrossRefGoogle Scholar
  2. Dehghan M, Hajarian M (2008) An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation. Appl Math Comput 202:571–588MathSciNetzbMATHGoogle Scholar
  3. Dehghan M, Hajarian M (2009) Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation A 1 X 1 B 1 + A 2 X 2 B 2 = C. Math Comput Model 49:1937–1959CrossRefGoogle Scholar
  4. Dehghan M, Hajarian M (2010a) The reflexive and anti-reflexive solutions of a linear matrix equation and systems of matrix equations. Rocky Mt J Math 40:825–848MathSciNetCrossRefGoogle Scholar
  5. Dehghan M, Hajarian M (2010b) On the reflexive and anti-reflexive solutions of the generalized coupled Sylvester matrix equations. Int J Syst Sci 41:607–625CrossRefGoogle Scholar
  6. Dehghan M, Hajarian M (2012a) On the generalized reflexive and anti-reflexive solutions to a system of matrix equations. Linear Algebra Appl 437:2793–2812MathSciNetCrossRefGoogle Scholar
  7. Dehghan M, Hajarian M (2012b) Two iterative algorithms for solving coupled matrix equations over reflexive and anti-reflexive matrices. Comput Appl Math 31:353–371MathSciNetCrossRefGoogle Scholar
  8. Dehghan M, Shirilord A (2019) A generalized modified Hermitian and skew-Hermitian splitting (GMHSS) method for solving complex Sylvester matrix equation. Appl Math Comput 348:632–651MathSciNetGoogle Scholar
  9. Golub GH, Van Loan CF (1996) Matrix computations, 3rd edn. The Johns Hopkins University Press, BaltimorezbMATHGoogle Scholar
  10. Horn RA, Johnson CR (1991) Topics in matrix analysis. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  11. Li F-L, Hu X-Y, Zhang L (2008) The generalized anti-reflexive solutions for a class of matrix equations (BX = C, XD = E). Appl Math Comput 27:31–46MathSciNetzbMATHGoogle Scholar
  12. Peng XY, Hu XY, Zhang L (2005) An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB = C. Appl Math Comput 160:763–777MathSciNetzbMATHGoogle Scholar
  13. Peng YX, Hu XY, Zhang L (2007) The reflexive and anti-reflexive solutions of the matrix equation A H XB = C. Appl Math Comput 186:638–645MathSciNetGoogle Scholar
  14. Ramadan MA, El-Danaf TS, Bayoumi AME (2015) Two iterative algorithms for the reflexive and Hermitian reflexive solutions of the generalized Sylvester matrix equation. J Vib Control 21:483–492MathSciNetCrossRefGoogle Scholar
  15. Zhan JC, Zhou SZ, Hu XY (2009) The (P, Q) generalized reflexive and anti-reflexive solutions of the matrix equation AX = B. Appl Math Comput 209:254–258MathSciNetzbMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  • Mohamed A. Ramadan
    • 1
  • Naglaa M. El-Shazly
    • 1
    Email author
  • Basem I. Selim
    • 1
  1. 1.Department of Mathematics and Computer Science, Faculty of ScienceMenoufia UniversityShebein El-KoomEgypt

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