Construction of Multivector Inverse for Clifford Algebras Over \(2m+1\)-Dimensional Vector Spaces from Multivector Inverse for Clifford Algebras Over 2m-Dimensional Vector Spaces

  • Eckhard HitzerEmail author
  • Stephen J. Sangwine
Part of the following topical collections:
  1. Proceedings ICCA 11, Ghent, 2017


Assuming known algebraic expressions for multivector inverses in any Clifford algebra over an even dimensional vector space \(\mathbb {R}^{p',q'}\), \(n'=p'+q'=2m\), we derive a closed algebraic expression for the multivector inverse over vector spaces one dimension higher, namely over \(\mathbb {R}^{p,q}\), \(n=p+q=p'+q'+1=2m+1\). Explicit examples are provided for dimensions \(n'=2,4,6\), and the resulting inverses for \(n=n'+1=3,5,7\). The general result for \(n=7\) appears to be the first ever reported closed algebraic expression for a multivector inverse in Clifford algebras Cl(pq), \(n=p+q=7\), only involving a single addition of multivector products in forming the determinant.


Clifford algebra Multivector determinants Multivector inverse 

Mathematics Subject Classification

Primary 15A66 Secondary 11E88 15A15 15A09 



The authors thank the organizers of the ICCA11 conference, and the anonymous reviewers for their helpful suggestions. E.H. wishes to thank God – Soli Deo Gloria, and his dear family for their steady support.


  1. 1.
    Abłamowicz, R., Fauser, B.: CLIFFORD - A Maple 17 Package for Clifford Algebra Computations,
  2. 2.
    Acus, A., Dargys, A.: Inverse of Multivector: Beyond \(p+q=5\) Threshold. Adv. Appl. Clifford Algebras 28, 65 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dadbeh, P.: Inverse and Determinant in \(0\) to \(5\) Dimensional Clifford Algebras. arXiv: 1104.0067 (2011)
  4. 4.
    Hitzer, E.: Introduction to Clifford’s Geometric Algebra, SICE Journal of Control, Measurement, and System Integration 514:338–350 (2012). arxiv:1306.1660
  5. 5.
  6. 6.
    Hitzer, E., Sangwine, S.: Multivector and multivector matrix inverses in real Clifford algebras. Appl. Math. Comput. 311, 375–389 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Mayer-Rüth, O.: (ARD Istanbul), “Für schmutzige Deals stehe ich nicht zur Verfügung” (I am not available for dirty deals). Accessed 20 Jan 2018
  8. 8.
    Lounesto, P.: Clifford Algebras and Spinors, London Mathematical Society Lecture Note Series (Book 286), 2nd edn. Cambridge University Press, Cambridge (2001)Google Scholar
  9. 9.
    Lundholm, D.: Geometric (Clifford) algebra, and applications, M.Sc. Thesis (2006), Department of Mathematics, Royal Institute of Technology, Sweden. arXiv:math/0605280 [math.RA]
  10. 10.
    Lundholm, D., Svensson, L.: Clifford algebra, geometric algebra, and applications (2016). arXiv:0907.5356v1 [math-ph].
  11. 11.
    Sangwine, S., Hitzer, E.: Clifford Multivector Toolbox (for MATLAB). Adv. in App. Cliff. Algs. 27(1), 539–558 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Shirokov, D.S.: Concepts of trace, determinant and inverse of Clifford algebra elements. In: Proc. 8th Congress of ISAAC, edited by V. I. Burenkov et al, Vol. 1, pp. 187–194, Friendship Univ. of Russia (2012). arXiv: 1108.5447

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.International Christian UniversityMitaka-shiJapan
  2. 2.School of Computer Science and Electronic EngineeringUniversity of EssexColchesterUK

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