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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
Article
  • 27 Downloads
Part of the following topical collections:
  1. Proceedings ICCA 11, Ghent, 2017

Abstract

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.

Keywords

Clifford algebra Multivector determinants Multivector inverse 

Mathematics Subject Classification

Primary 15A66 Secondary 11E88 15A15 15A09 

Notes

Acknowledgments

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.

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Copyright information

© 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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