An Algebraic Model for Real Matrix Representations. Remarks Regarding Quaternions and Octonions

  • Cristina FlautEmail author
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 179)


In this chapter, we present some applications of quaternions and octonions. We present the real matrix representations for complex octonions and some of their properties which can be used in computations where these elements are involved. Moreover, we give a set of invertible elements in split quaternion algebras and in split octonion algebras.


Quaternion algebras Octonion algebras Matrix representation 

2000 AMS Subject Classification

17A35 15A06 15A24 16G30 


  1. Alamouti, S.M.: A simple transmit diversity technique for wireless communications. IEEE J. Sel. Areas Commun. 16(8), 1451–1458 (1998)CrossRefGoogle Scholar
  2. Alfsmann, D., Göckler, H.G., Sangwine, S.J., Ell, T., A.: Hypercomplex algebras in digital signal processing: benefits and drawbacks. In: 15th European Signal Processing Conference (EUSIPCO, 2007), Poznan, Poland, pp. 1322–1326 (2007)Google Scholar
  3. Baez, J.C.: The octonions. B. Am. Math. Soc. 39(2), 145–205 (2002)CrossRefGoogle Scholar
  4. Chanyal, B. C.: Octonion massive electrodynamics. Gen. Relat. Gravit. 46, article ID: 1646 (2014)Google Scholar
  5. Chanyal, B.C., Bisht, P.S., Negi, O.P.S.: Generalized split-octonion electrodynamics. Int. J. Theor. Phys. 50(6), 1919–1926 (2011)MathSciNetCrossRefGoogle Scholar
  6. Chen, J., Tu, A.: Fabric image edge detection based on octonion and echo state networks. Appl. Mech. Mater. 263–266, 2483–2487 (2013)CrossRefGoogle Scholar
  7. Flaut, C., Shpakivskyi, V.: Real matrix representations for the complex quaternions. Adv. Appl. Clifford Algebras 23(3), 657–671 (2013)MathSciNetCrossRefGoogle Scholar
  8. Hanson, A.J.: Visualizing Quaternions. Elsevier Morgan Kaufmann Publishers, Burlington (2006)Google Scholar
  9. Jia, Y.B.: Quaternion and Rotation, Com S 477/577 Notes, (2017)Google Scholar
  10. Jouget, P.: Sécurité et performance de dispositifs de distribution quantique de clés à variables continues. Ph.D Thesis, TELECOM ParisTech (2013)Google Scholar
  11. Klco, P., Smetana, M., Kollarik, M., Tatar, M.: Application of octonions in the cough sounds classification. Adv. Appl. Sci. Res. 8(2), 30–37 (2017)Google Scholar
  12. Kostrikin, A.I., Shafarevich, I.R. (eds.): Algebra VI. Springer, Berlin (1995)Google Scholar
  13. Li, X.M.: Hyper-Complex Numbers and its Applications in Digital Image Processing. Seminars and Distinguished Lectures (2011)Google Scholar
  14. Schafer, R.D.: An Introduction to Nonassociative Algebras. Academic, New York (1966)zbMATHGoogle Scholar
  15. Snopek, K., M.: Quaternions and Octonions in Signal Processing - Fundamentals and Some New Results, Przeglad Telekomunikacyjny - Wiadomoś ci Telekomunikacyjne, SIGMA NOT, 134(6), 619–622 (2015)CrossRefGoogle Scholar
  16. Tian, Y.: Matrix representations of octonions and their applications. Adv. in Appl. Clifford Algebras 10(1), 61–90 (2000)MathSciNetCrossRefGoogle Scholar
  17. Unger, T., Markin, N.: Quadratic forms and space-time block codes from generalized quaternion and biquaternion algebras. IEEE Trans. Inf. Theory 57(9), 6148–6156 (2011)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceOvidius UniversityConstantaRomania

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