K. S. Abdulkhaev | Searcharxiv

arXiv2026

Explicit Formula for Inverse and Determinant in Geometric Algebras over Odd-dimensional Vector Spaces

In this paper, we present explicit formulas for the inverse and determinant in geometric (Clifford) algebras over vector spaces of dimension $n=7$. The derivation of these formulas is made possible by generalizing the concept of conjugation to basis conjugation operations. We further develop a general method for constructing such formulas over odd-dimensional spaces from the known even-dimensional case. To validate computational utility of the results, we provide a numerical implementation of the formulas. The code implementation is available at the repository github.com/kamranuz/clifford_7d. These formulas extend previous results for lower dimensions and offer new insights for applications in mathematical physics and computational geometry.

math.RA↗

arXiv2022

Basis-free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras

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.

math-ph↗