On Explicit Formulas for Characteristic Polynomial Coefficients in Geometric Algebras

Kamron Abdulkhaev, Dmitry Shirokov

Abstract: In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras \(\mathcal {G}_{p,q}\) of vector space of dimension \(n=p+q\). For the first time, we present explicit formulas for all characteristic polynomial coefficients in the case \(n=5\). The formulas involve only the operations of geometric product, summation, and operations of conjugation. For the first time, we present an analytic proof of the corresponding formulas in the case \(n=4\). We present some new properties of the operations of conjugation and grade projection and use them to obtain the main results of this paper. The results of this paper can be used in different applications of geometric algebras in computer graphics, computer vision, engineering, and physics. The presented explicit formulas for characteristic polynomial coefficients can also be used in symbolic computation.