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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

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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

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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