Spectral square roots of the multivector

TL;DR

This paper introduces a method for computing multiple spectral square roots of multivectors in real Clifford algebras, leveraging Bott's periodicity and eigensystem analysis, applicable to both numerical and symbolic coefficients.

Contribution

It presents a novel approach combining Bott's periodicity and eigensystem analysis to find multivector square roots in Clifford algebras, including symbolic cases and domain determination.

Findings

Method successfully computes multivector square roots in low and high-dimensional Clifford algebras.

Examples include 4D Euclidean, anti-Euclidean, and relativistic Clifford algebras.

Tables of basis vectors for matrix conversion are provided for real Clifford algebras.

Abstract

The problem of multivector (MV) multiple square roots in real geometric Clifford algebras Cl(p,q) with symbolic coefficients is considered. The method to find multiple MV square roots that is based on R.Bott's periodicity table and matrix eigensystem in Cl(p,q) is proposed. The method can be applied to MV having both numerical and symbolic coefficients. In addition, method allows to determine the domain of the existence of thus obtained spectral square roots. A number of examples is presented for multivectors in low, p+q<= 3, and higher dimensional Clifford algebras, including 4D (anti)-Euclidean space and relativistic Cl(1,3) and Cl(3,1) algebras. Tables of the required basis vectors for conversion of MV to Bott's matrix representation have been found from respective algebra idempotents using ideal theory and presented for real Clifford algebras in Appendix.