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Spectral square roots of the multivector

Adolfas Dargys, Arturas Acus

TL;DR

This work introduces a spectral diagonalization method for computing multivector square roots in real Clifford algebras $Cl_{p,q}$ by mapping MV elements to Bott-based matrix reps, performing eigen-decomposition, and rebuilding MV roots from signed diagonal roots. The approach applies to both symbolic and numeric MV coefficients and includes explicit domain conditions for the existence of roots, with comprehensive demonstrations across 1–4D algebras and relativistic cases ($Cl_{1,3}$, $Cl_{3,1}$). Key findings include detailed closed-form root expressions in low dimensions, the emergence of sign-paired root families, and the identification of how algebra isomorphisms can mislead root counting for certain spinors. The method is supplemented by numerical diagonalization for higher dimensions and by a quadratic MV equation treatment, offering a practical framework for applications in spinor theory, Clifford-Fourier transforms, and control/robotics where symbolic MV roots are advantageous.

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.

Spectral square roots of the multivector

TL;DR

This work introduces a spectral diagonalization method for computing multivector square roots in real Clifford algebras by mapping MV elements to Bott-based matrix reps, performing eigen-decomposition, and rebuilding MV roots from signed diagonal roots. The approach applies to both symbolic and numeric MV coefficients and includes explicit domain conditions for the existence of roots, with comprehensive demonstrations across 1–4D algebras and relativistic cases (, ). Key findings include detailed closed-form root expressions in low dimensions, the emergence of sign-paired root families, and the identification of how algebra isomorphisms can mislead root counting for certain spinors. The method is supplemented by numerical diagonalization for higher dimensions and by a quadratic MV equation treatment, offering a practical framework for applications in spinor theory, Clifford-Fourier transforms, and control/robotics where symbolic MV roots are advantageous.

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.

Paper Structure

This paper contains 32 sections, 124 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Multivector matrix reps
  3. Diagonalization method
  4. Square root algorithm
  5. Some practical issues
  6. Eigensystem and related problems
  7. The square root of idempotent
  8. Spectral MV roots in low dimensional CAs
  9. MV roots in 1D algebras
  10. $\mathit{Cl}_{0,1}$ algebra, $\mathbb{C}(1)$ (complex numbers)
  11. $\mathit{Cl}_{1,0}$ algebra, ${^2\mathbb{R}(1)}$ (hyperbolic numbers)
  12. MV roots in 2D algebras
  13. $\mathit{Cl}_{0,2}$ algebra, $\mathbb{H}(1)$. (Hamilton quaternions)
  14. $\mathit{Cl}_{1,1}$ algebra, $\mathbb{R}(2)$
  15. $\mathit{Cl}_{2,0}$ algebra, $\mathbb{R}(2)$
  16. ...and 17 more sections

Figures (2)

  • Figure 1: Complex plane illustration of sum of square roots $s$ and $s^*$ with complex conjugate coefficients inside the roots that corresponds to real multivector $(s+s^{*})A_r$, where $A_r$ may be, for example, an elementary blade. a) $s+s^{*}=\sqrt{1+\mathrm{i}}+\sqrt{1-\mathrm{i}}=2^{5/4}\cos(\pi/8)$. b) $s+s^{*}=\sqrt{-1+\mathrm{i}}+\sqrt{-1-\mathrm{i}}=2^{5/4}\cos(3\pi/8)$.
  • Figure 2: Analogy between unit circle ($x^2+y^2=r^2$, where $r$ is a circle radius) and rectangular hyperbola ($x^2-y^2=r^2$, where $r=\text{const}$ is a hyperbolic radius, which is a distance on the horizontal axis between coordinate center and intersection of hyperbola with the horizontal axes). For $r=1$: a) rotation of a MV $\mathsf{A}$ in a complex plane and b) boost of a MV $\mathsf{A}$ in a hyperbolic plane. The end of radius-vector $\mathsf{A}=a_0+a_1\mathbf{e}_{1}$ moves along circle or rectangular hyperbola and represents unit MV: $\lvert \mathsf{A}\rvert=\sqrt{a_0^2+a_1^2}=1$ for circle when $\mathbf{e}_{1}^2=-1$, and $\lvert \mathsf{A}\rvert=\sqrt{a_0^2-a_1^2}=1$ for hyperbola when $\mathbf{e}_{1}^2=1$. The coordinates of $\mathsf{A}$ are $a_0=\cos\varphi_t$and $a_1=\sin\varphi_t$ for circle, and $a_0=\cosh\varphi_h$ and $a_1=\sinh\varphi_h$ for hyperbola.