Hitzer & Sangwine: Technical Report on Multivector and multivector matrix inverses in real Clifford algebras


Multivector and multivector matrix inverses in real Clifford algebras
by Eckhard Hitzer, College of Liberal Arts, International Christian University, 181-8585 Mitaka, Tokyo, Japan. hitzer_AT_icu.ac.jp;
Stephen Sangwine, School of Computer Science and Electronic Engineering, University of Essex, Colchester, CO4 3SQ, UK. sjs_AT_essex.ac.uk

School of Computer Science and Electronic Engineering, University of Essex, UK,
Technical Report CES-534, ISSN: 1744-8050
Download (PDF): http://repository.essex.ac.uk/17282

Abstract We show how to compute the inverse of multivectors in finite dimensional real Clifford algebras Cl(p; q). For algebras over vector spaces of fewer than six dimensions, we provide explicit formulae for discriminating between divisors of zero and invertible multivectors, and for the computation of the inverse of a general invertible multivector. For algebras over vector spaces of dimension six or higher, we use isomorphisms between algebras, and between multivectors and matrix representations with multivector elements in Clifford algebras of lower dimension. Towards this end we provide explicit details of how to compute several forms of isomorphism that are essential to invert multivectors in arbitrarily chosen algebras. We also discuss briefly the computation of the inverses of matrices of multivectors by adapting an existing textbook algorithm for matrices to the multivector setting, using the previous results to compute the required inverses of individual multivectors.
Keywords: Cliord algebra, inverse, multivector matrix algebra
2000 MSC: 15A66, 11E88, 15A09, 68-04

Source: http://repository.essex.ac.uk/17282, 22nd of July 2016.

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