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The generalised Sylvester matrix equations over the generalised bisymmetric and skew-symmetric matrices

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  • Mehdi Dehghan
  • Masoud Hajarian

Abstract

A matrix P is called a symmetric orthogonal if P = PT = P−1. A matrix X is said to be a generalised bisymmetric with respect to P if X = XT = PXP. It is obvious that any symmetric matrix is also a generalised bisymmetric matrix with respect to I (identity matrix). By extending the idea of the Jacobi and the Gauss–Seidel iterations, this article proposes two new iterative methods, respectively, for computing the generalised bisymmetric (containing symmetric solution as a special case) and skew-symmetric solutions of the generalised Sylvester matrix equation (including Sylvester and Lyapunov matrix equations as special cases) which is encountered in many systems and control applications. When the generalised Sylvester matrix equation has a unique generalised bisymmetric (skew-symmetric) solution, the first (second) iterative method converges to the generalised bisymmetric (skew-symmetric) solution of this matrix equation for any initial generalised bisymmetric (skew-symmetric) matrix. Finally, some numerical results are given to illustrate the effect of the theoretical results.

Suggested Citation

  • Mehdi Dehghan & Masoud Hajarian, 2012. "The generalised Sylvester matrix equations over the generalised bisymmetric and skew-symmetric matrices," International Journal of Systems Science, Taylor & Francis Journals, vol. 43(8), pages 1580-1590.
    • Handle: RePEc:taf:tsysxx:v:43:y:2012:i:8:p:1580-1590
    DOI: 10.1080/00207721.2010.549584
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    Cited by:

    1. Hu, Jingjing & Ma, Changfeng, 2018. "Conjugate gradient least squares algorithm for solving the generalized coupled Sylvester-conjugate matrix equations," Applied Mathematics and Computation, Elsevier, vol. 334(C), pages 174-191.
    2. Xie, Ya-Jun & Ma, Chang-Feng, 2015. "The MGPBiCG method for solving the generalized coupled Sylvester-conjugate matrix equations," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 68-78.
    3. repec:taf:tsysxx:v:46:y:2015:i:3:p:488-502 is not listed on IDEAS
    4. Huang, Baohua & Ma, Changfeng, 2018. "An iterative algorithm for the least Frobenius norm least squares solution of a class of generalized coupled Sylvester-transpose linear matrix equations," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 58-74.
    5. Baohua Huang & Changfeng Ma, 2019. "The least squares solution of a class of generalized Sylvester-transpose matrix equations with the norm inequality constraint," Journal of Global Optimization, Springer, vol. 73(1), pages 193-221, January.

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