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

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## 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.

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