[Paper Review] Inverse of multivector: Beyond p+q=5 threshold

A. Acus, A. Dargys|arXiv (Cornell University)|Dec 14, 2017

Algebraic and Geometric Analysis6 references6 citations

TL;DR

This paper extends the grade-negation method for computing the inverse of multivectors in Clifford geometric algebra beyond the previously established $ n = 5 $ threshold, presenting explicit, coordinate-free formulas for $ n = 6 $ using linear combinations of grade-negated geometric products. The key contribution is a systematic algorithm that generates compact inverse expressions independent of signature $ (p,q) $, enabling symbolic and numerical computation in higher-dimensional geometric algebras.

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 $n\le5$.

Motivation & Objective

• To overcome the longstanding limitation of existing multivector inverse formulas, which were only valid up to $ n = 5 $, by extending the grade-negation method to higher dimensions.
• To develop a general algorithm for computing the inverse of a general multivector in $ \mathrm{Cl}_{p,q} $ for $ n = p+q = 6 $, applicable across all signatures.
• To provide explicit, compact, and coordinate-free inverse formulas that reduce computational complexity and avoid reliance on matrix representations.
• To reveal structural patterns in determinant norms and inverse expressions, suggesting potential for extension to $ n > 6 $.
• To demonstrate that the inverse of a multivector in $ n=6 $ can generate inverse formulas for all lower-dimensional algebras ($ n \leq 6 $) via a unified framework.

Proposed method

• The method generalizes the grade-negation approach by introducing linear combinations of geometric products of a multivector and its grade-negated counterparts, rather than relying on single products.
• It uses the geometric product of the multivector with multiple grade-negated versions to construct scalar-valued determinant norms, which are then used to derive the inverse.
• The algorithm is built on the principle that the product of a multivector with carefully selected grade-negated forms yields a scalar, analogous to a determinant, enabling inverse extraction.
• The authors employ a systematic construction of self-negated products and analyze their structure to identify minimal and symmetric forms, including a three-term symmetric formula with equal weights.
• They leverage the 8-periodicity of Clifford algebras to predict the number of required multivector products in the norm construction, guiding the search for compact expressions.
• The method avoids matrix representations entirely, relying only on geometric products, grade projections, and negation operations, ensuring coordinate-free and signature-independent results.

Experimental results

Research questions

• RQ1 Can the grade-negation method for multivector inversion be extended beyond the $ n=5 $ limit, and if so, what structural modifications are required?
• RQ2 What is the minimal and most compact form of the inverse multivector expression for $ n=6 $, and how can it be expressed as a linear combination of grade-negated products?
• RQ3 Do the determinant norm formulas for $ n=6 $ encapsulate all lower-dimensional cases ($ n \leq 5 $), and if so, how?
• RQ4 Can symmetric, balanced formulas (e.g., three-term linear combinations with equal weights) be constructed for $ n=6 $, and what is their computational significance?
• RQ5 Is there a universal pattern or structure—such as an 'onion-like' hierarchy—in the determinant norms across different dimensions, suggesting a generalizable formula for all $ n $?

Key findings

• The paper presents the first explicit, closed-form inverse formulas for general multivectors in all $ \mathrm{Cl}_{p,q} $ algebras of dimension $ n=6 $, valid for all signatures.
• These formulas are constructed as linear combinations of two or three grade-negated geometric products, with the two-product form being computationally optimal due to reduced multiplication count.
• The inverse formulas for $ n=6 $ naturally reduce to known $ n \leq 5 $ expressions, demonstrating a hierarchical, 'onion-like' structure where higher-dimensional formulas subsume lower ones.
• A symmetric three-term formula with equal weights $ \frac{1}{3} $ is derived, though less efficient computationally, it may inspire generalizations to higher dimensions.
• The method avoids matrix representations entirely, relying only on geometric products and grade negations, ensuring coordinate-free and signature-independent computation.
• The authors conjecture that such formulas exist for all dimensions $ n $, based on observed structural patterns and the 8-periodicity of Clifford algebras.