\({2\times 2}\) Matrix Representation Forms and Inner Relationships of Split Quaternions

  • Qiu-Ying Ni
  • Jin-Kou Ding
  • Xue-Han ChengEmail author
  • Ying-Nan Jiao


This paper proposes a \({2\times 2}\) real matrix isomorphic representation form of split quaternions and a \({2m\times 2n}\) real matrix isomorphic representation form of \({m\times n}\) split quaternion matrices. In particular, we highlighted the inner relationships among the existing different \({2\times 2}\) matrix representation forms of split quaternions. Furthermore, we studied the inner relationships among different \({2\times 2}\) matrix representation forms of Hamilton quaternions as an application. The forms and relationships discussed in this paper can simplify the computation and make split quaternions get extensive applications.


Split quaternions Isomorphic Inner relationships Hamilton quaternions 

Mathematics Subject Classification




  1. 1.
    Antonuccio, F.: Split-quaternions and the Dirac equation. Adv. Appl. Clifford Algebras. 25(1), 13–29 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ata, E., Yayli, Y.: Split quaternions and semi-Euclidean projective spaces. Chaos Solitons Fractals 41(4), 1910–1915 (2009)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Brody, D.C., Graefe, E.-M.: On complexified mechanics and coquaternions. J. Phys. A Math. Theor. 44(7), 072001 (2011)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Cockle, J.L.I.I.: On systems of algebra involving more than one imaginary; and on equations of the fifth degree. Lond. Edinb. Dublin Philos. Mag. J. Sci. 35(238), 434–437 (1849)CrossRefGoogle Scholar
  5. 5.
    Erdoğdu, M., Özdemir, M.: On eigenvalues of split quaternion matrices. Adv. Appl. Clifford Algebras 23(3), 615–623 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Erdoğdu, M., Özdemir, M.: On exponential of split quaternionic matrices. Appl. Math. Comput. 315, 468–476 (2017)MathSciNetGoogle Scholar
  7. 7.
    Falcão, M., Irene, M.F., Severino, R.: On the roots of coquaternions. Adv. Appl. Clifford Algebras 28(5), 97 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kula, L., Yayli, Y.: Split quaternions and rotations in semi Euclidean space E 4 2. J. Korean Math. Soc. 44(6), 1313–1327 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Guo, A., Salamo, G., Duchesne, D.: Observation of P T-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103(9), 093902 (2009)ADSCrossRefGoogle Scholar
  10. 10.
    Özdemir, M.: The roots of a split quaternion. Appl. Math. Lett. 22(2), 258–263 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Pop, M.-I., Cretu, N.: Intrinsic transfer matrix method and split quaternion formalism for multilayer media. Wave Motion 65, 105–111 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ujang, B.C., Took, C.C., Mandic, D.P.: Split quaternion nonlinear adaptive filtering. Neural Netw. 23(3), 426–434 (2010)CrossRefGoogle Scholar
  13. 13.
    Zhang, Z., Jiang, Z., Jiang, T.: Algebraic methods for least squares problem in split quaternionic mechanics. Appl. Math. Comput. 269, 618–625 (2015)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Zhao, K., Schaden, M., Wu, Z.: Enhanced magnetic resonance signal of spin-polarized Rb atoms near surfaces of coated cells. Phys. Rev. A 81(4), 042903 (2010)ADSCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Qiu-Ying Ni
    • 1
    • 2
  • Jin-Kou Ding
    • 1
  • Xue-Han Cheng
    • 3
    Email author
  • Ying-Nan Jiao
    • 4
  1. 1.School of ScienceBeijing University of Posts and TelecommunicationsBeijingChina
  2. 2.State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and TelecommunicationsBeijingChina
  3. 3.School of Mathematics and Statistics ScienceLudong UniversityYantaiChina
  4. 4.National Computer Network and information Security Administrative CenterBeijingChina

Personalised recommendations