# Acus & Dargys: Inverse of multivector: Beyond p+q=5 threshold

A. Acus, A. Dargys, Inverse of multivector: Beyond p+q=5 threshold, Submitted on 14 Dec 2017, https://arxiv.org/abs/1712.05204

Abstract: The algorithm of finding inverse multivector (MV) in a symbolic form is of paramount importance in computational Clifford or geometric algebra (GA) Clp,q. The first attempts of inversion of general MV were based on matrix representation of MV basis elements. However, the complexity of such calculations grows exponentially with the dimension n=p+q of Clp,q algebra. The breakthrough occurred 10 years later (P. Dadbeh, 2011), after grade-negation operation was introduced. It has allowed to write down explicit and compact inverse MVs as a product of initial MV and its carefully chosen grade-negation counterparts for all GAs up to dimension n5. In this report we show that the grade-negation self-product method can be extended beyond p+q=5 threshold if, in addition, properly constructed multilinear combinations of such MV products are used. In particular, we write down compact and explicit MV inverse formulas for all p+q=6 algebras. For readers convenience we have also presented inverse MVs for lower algebras, p+q5, in a form of grade negations.