## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 20, Number 4 (2013), 639-653.

### The reflexive and Hermitian reflexive solutions of the generalized Sylvester-conjugate matrix equation

Masoud Hajarian and Mehdi Dehghan

#### Abstract

The main purpose of this correspondence is to establish two gradient based iterative (GI) methods extending the Jacobi and Gauss-Seidel iterations for solving the generalized Sylvester-conjugate matrix equation \begin{equation*} A_1XB_1+A_2\overline{X}B_2+C_1YD_1+C_2\overline{Y}D_2=E, \end{equation*} over reflexive and Hermitian reflexive matrices. It is shown that the iterative methods, respectively, converge to the reflexive and Hermitian reflexive solutions for any initial reflexive and Hermitian reflexive matrices. We report numerical tests to show the effectiveness of the proposed approaches.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 4 (2013), 639-653.

**Dates**

First available in Project Euclid: 22 October 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1382448185

**Digital Object Identifier**

doi:10.36045/bbms/1382448185

**Mathematical Reviews number (MathSciNet)**

MR3129064

**Zentralblatt MATH identifier**

1282.15014

**Subjects**

Primary: 15A24: Matrix equations and identities 65F10: Iterative methods for linear systems [See also 65N22] 65F30: Other matrix algorithms

**Keywords**

The generalized Sylvester-conjugate matrix
equations Reflexive solution pair Hermitian reflexive solution pair Iterative method

#### Citation

Hajarian, Masoud; Dehghan, Mehdi. The reflexive and Hermitian reflexive solutions of the generalized Sylvester-conjugate matrix equation. Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 4, 639--653. doi:10.36045/bbms/1382448185. https://projecteuclid.org/euclid.bbms/1382448185