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.
The generalized Sylvester equation Finite iterative method Anti-reflexive matrices The least Frobenius norm Optimal approximation solution
Mathematics Subject Classification
65F10 65F30 15A06
This is a preview of subscription content, log in to check access.
The authors are grateful to the referees for their valuable suggestions and comments for the improvement of the paper.
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
Dehghan M, Hajarian M (2009) Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation A1X1B1 + A2X2B2 = C. Math Comput Model 49:1937–1959CrossRefGoogle Scholar
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
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
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
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
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
Golub GH, Van Loan CF (1996) Matrix computations, 3rd edn. The Johns Hopkins University Press, BaltimorezbMATHGoogle Scholar
Horn RA, Johnson CR (1991) Topics in matrix analysis. Cambridge University Press, CambridgeCrossRefGoogle Scholar
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
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
Peng YX, Hu XY, Zhang L (2007) The reflexive and anti-reflexive solutions of the matrix equation AHXB = C. Appl Math Comput 186:638–645MathSciNetGoogle Scholar
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
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