Inverse of multivector: Beyond p+q=5 threshold

TL;DR

This paper extends the methods for finding inverse multivectors in Clifford geometric algebra beyond the previously established dimension limit of 5, providing explicit formulas for dimension 6.

Contribution

It introduces a new approach using linear combinations of grade-negation products to compute inverses in higher dimensions, surpassing the previous threshold of n=5.

Findings

Explicit inverse formulas for n=6 Clifford algebras.

Extension of grade-negation methods beyond n=5.

Unified formulas encompassing lower dimensions.

Abstract

The algorithm of finding inverse multivector (MV) numerically and symbolically is of paramount importance in the applied Clifford geometric algebra (GA) $Cl_{p,q}$. The first general MV inversion algorithm was based on matrix representation of MV. The complexity of calculations and size of the answer in a symbolic form grow exponentially with the GA dimension $n=p+q$. The breakthrough occurred when D. Lundholm and then P. Dadbeh found compact inverse formulas up to dimension $n\le5$. The formulas were constructed in a form of Clifford product of initial MV and its carefully chosen grade-negation counterparts. In this report we show that the grade-negation self-product method can be extended beyond $n=5$ threshold if, in addition, properly constructed linear combinations of such MV products are used. In particular, we present compact explicit MV inverse formulas for algebras of vector space dimension $n=6$ and show that they embrace all lower dimensional cases as well. For readers convenience, we have also given various MV formulas in a form of grade negations when $n\le5$.