Explicit Formula for Inverse and Determinant in Geometric Algebras over Odd-dimensional Vector Spaces
2026-06-22 • Symbolic Computation
Symbolic Computation
AI summaryⓘ
The authors explain how to find special formulas that let you calculate the inverse and determinant in 7-dimensional geometric (Clifford) algebras, which are mathematical systems used in physics and geometry. They extended the idea of conjugation by introducing new ‘basis conjugation’ operations to help create these formulas. The authors also describe a general way to build these formulas for odd dimensions using what’s already known for even dimensions. To show these formulas work well, they include a computer code for calculations. This work builds on earlier results for lower dimensions and could help in fields like mathematical physics.
Geometric algebraClifford algebraInverseDeterminantConjugationBasis conjugationOdd-dimensional spacesEven-dimensional spacesMathematical physicsComputational geometry
Authors
K. S. Abdulkhaev, D. S. Shirokov
Abstract
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.