Skip to main content
SpringerLink
Log in

On solutions of matrix equation XF−AX=C and \(XF-A\widetilde{X}=C\) over quaternion field

  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

By means of Kronecker map and complex representation of a quaternion matrix, some explicit solutions to the quaternion matrix equations XF−AX=C and \(XF-A\widetilde{X}=C\) are established. One of the solutions is neatly expressed by a symmetric matrix, a controllability matrix and an observability matrix. In addition, two practical algorithms for these two equations are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kaiser, H., George, E.A., Werner, S.A.: Neutron interferometrix search for quaternions in quantum mechanics. Phys. Rev. A 29, 2276–2279 (1984)

    Article  Google Scholar 

  2. Klein, A.G.: Schrödinger inviolate: neutron optical searches for violations of quatum mechanics. Physica B 151, 44–49 (1988)

    Article  Google Scholar 

  3. Daves, A.J., Mckellar, B.H.: Non-relativistic quaternionic quantum mechanics. Phys. Rev. A 40, 4209–4214 (1989)

    Article  MathSciNet  Google Scholar 

  4. Daves, A.J.: Quaternionic Dirac equation. Phys. Rev. D 41, 2628–2630 (1990)

    Article  MathSciNet  Google Scholar 

  5. Daves, A.J., Mckellar, B.H.: Observability of quaternionic quantum mechanics. Phys. Rev. A 46, 3671–3675 (1992)

    Article  Google Scholar 

  6. Adler, S.L.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford Univ. Press, New York (1994)

    Google Scholar 

  7. Paige, C.C., Saunders, A.: LSQR: An algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Softw. 8, 43–71 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  8. Peres, A.: Proposed test for complex versus quaternion quantum theory. Phys. Rev. Lett. 42, 683–686 (1979)

    Article  Google Scholar 

  9. Jiang, T., Chen, L.: Algebraic algorithms for least squares problem in quaternionic quantum theory. Comput. Phys. Commun. 176, 481–485 (2007)

    Article  MATH  Google Scholar 

  10. Jiang, T., Chen, L.: An algebraic method for Schrödinger equation in quaternionic quantum mechanics. Comput. Phys. Commun. 178, 795–799 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jiang, T., Zhao, J., Wei, M.: A new technique of quaternion equality constrained least squares problem. J. Comput. Appl. Math. 216, 509–513 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jiang, T., Wei, M.: Equality constrained least squares problem over quaternion field. Appl. Math. Lett. 16, 883–888 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Wang, M., Wei, M., Feng, Y.: An iterative algorithm for least squares problem in quaternionic quantum theory. Comput. Phys. Commun. 179, 203–207 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Huang, J.: Complex representation of the quaternion matrix equations and structure-preserving algorithm. Pure Appl. Math. 24, 321–326 (2008)

    MATH  Google Scholar 

  15. Huang, J.: The solution of the quaternion matrix equation AX−YB=C and its optimal approximation. Math. Theory Appl. 24, 1–4 (2004)

    Google Scholar 

  16. Huang, J.: Three categories solution on the quaternion matrix equation AX+YA=C. Nat. Sci. J. Harbin Norm. Univ. 20, 31–34 (2004)

    Google Scholar 

  17. Jiang, T., Wei, M.: On solution of the quaternion matrix equation \(X-A\widetilde{X}B=C\) and its application. Acta Math. Sin. 21, 483–490 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Huang, L.: The matrix equation AXB−CXD=E over the quaternion field. Linear Algebra Appl. 234, 197–208 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Duan, G.R.: Solutions to matrix equation AV+BW=VF and their application to eigenstructure assignment in linear systems. IEEE Trans. Automat. Control 38(2), 276–280 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bhattacharyya, S.P., De Souza, E.: Pole assignment via Sylvester’s equation. Syst. Control Lett. 1, 261–263 (1972)

    Article  Google Scholar 

  21. Duan, G.R.: On the solution to Sylvester matrix equation AV+BW=EVF. IEEE Trans. Automat. Control 41(4), 612–614 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Desouza, E., Bhattacharyya, S.P.: Controllability, observability and the solution of AX−XB=C. Linear Algebra Appl. 39, 167–188 (1981)

    Article  MathSciNet  Google Scholar 

  23. Wu, A., Duan, G.R.: On solution to the generalized Sylvester matrix equation AV+BW=EVF. IET Control Theory Appl. 1(1), 402–408 (2007)

    Article  MathSciNet  Google Scholar 

  24. Jiang, T.: A new algebra structure of quaternion. Acta Mech. Sin. 34, 116–122 (2002)

    Google Scholar 

  25. Hanzon, B., Peeters, R.M.: A Faddeev sequence method for solving Lyapunov and Sylvester equations. Linear Algebra Appl. 241–243, 401–430 (1996)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, East China Normal University, Shanghai, 200241, P.R. China

    Caiqin Song & Guoliang Chen

Authors
  1. Caiqin Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guoliang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guoliang Chen.

Additional information

This project is granted financial support from Shanghai Nature Science Foundation (no. 092R1408700), NSFC (no. 10901056), the Fundamental Research Funds for the Central Universities.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Song, C., Chen, G. On solutions of matrix equation XF−AX=C and \(XF-A\widetilde{X}=C\) over quaternion field. J. Appl. Math. Comput. 37, 57–68 (2011). https://doi.org/10.1007/s12190-010-0420-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-010-0420-9

Keywords

Mathematics Subject Classification (2000)

Search

Navigation