Basis-free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras

TL;DR

This paper derives basis-free formulas for characteristic polynomial coefficients in geometric algebras up to dimension six, verified by computer, with applications in graphics, vision, engineering, and physics.

Contribution

It introduces new basis-free formulas for characteristic polynomial coefficients in geometric algebras, including proofs and formulas for specific cases up to dimension five.

Findings

Formulas verified using computer calculations.

Analytical proof provided for n=4 and one for n=5.

Formulas for vectors and rotors in arbitrary and specific dimensions.

Abstract

In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras ${\mathcal {G}}_{p,q}$ of vector space of dimension $n=p+q$. We present basis-free formulas for all characteristic polynomial coefficients in the cases $n\leq 6$, alongside with a method to obtain general form of these formulas. The formulas involve only the operations of geometric product, summation, and operations of conjugation. All the formulas are verified using computer calculations. We present an analytical proof of all formulas in the case $n=4$, and one of the formulas in the case $n=5$. We present some new properties of the operations of conjugation and grade projection and use them to obtain the results of this paper. We also present formulas for characteristic polynomial coefficients in some special cases. In particular, the formulas for vectors (elements of grade $1$) and basis elements are presented in the case of arbitrary $n$, the formulas for rotors (elements of spin groups) are presented in the cases $n\leq 5$. The results of this paper can be used in different applications of geometric algebras in computer graphics, computer vision, engineering, and physics. The presented basis-free formulas for characteristic polynomial coefficients can also be used in symbolic computation.