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Outer and Eigen: Tangent Concepts

David Eelbode, Martin Roelfs, Steven De Keninck

TL;DR

This work reframes the invariant decomposition of bivectors through the outer exponential $\Lambda^B$, establishing a spectral picture via the adjoint action and deriving a Cayley–Hamilton–type factorization that operates as a square-root analogue for bivectors. It introduces outer trigonometric functions and the outer tangent $\mathop{\mathrm{t}}_\wedge(B)$, connecting them to a Cayley transform that maps bivectors to rotors and enables a constructive decomposition of $B$ into commuting simple bivectors. The paper shows that the spectrum $\sigma(B)$ and the invariant decomposition can be obtained from $\Lambda^B$ and the eigenvectors, with a robust framework that extends to pseudo-null bivectors and is linked to spinor structures. Overall, the outer exponential provides a unified toolset for encoding invariants, facilitating decomposition, and connecting geometric-algebraic transformations to classical matrix concepts. Applications to rotations, boosts, and higher-dimensional isoclinic cases are highlighted, with future work targeting Jordanesque behaviour and spinor interpretations.

Abstract

In this paper we use the power of the outer exponential $Λ^B$ of a bivector $B$ to see the so-called invariant decomposition from a different perspective. This is deeply connected with the eigenvalues for the adjoint action of $B$, a fact that allows a version of the Cayley-Hamilton theorem which factorises the classical theorem (both the matrix version and the geometric algebra version).

Outer and Eigen: Tangent Concepts

TL;DR

This work reframes the invariant decomposition of bivectors through the outer exponential , establishing a spectral picture via the adjoint action and deriving a Cayley–Hamilton–type factorization that operates as a square-root analogue for bivectors. It introduces outer trigonometric functions and the outer tangent , connecting them to a Cayley transform that maps bivectors to rotors and enables a constructive decomposition of into commuting simple bivectors. The paper shows that the spectrum and the invariant decomposition can be obtained from and the eigenvectors, with a robust framework that extends to pseudo-null bivectors and is linked to spinor structures. Overall, the outer exponential provides a unified toolset for encoding invariants, facilitating decomposition, and connecting geometric-algebraic transformations to classical matrix concepts. Applications to rotations, boosts, and higher-dimensional isoclinic cases are highlighted, with future work targeting Jordanesque behaviour and spinor interpretations.

Abstract

In this paper we use the power of the outer exponential of a bivector to see the so-called invariant decomposition from a different perspective. This is deeply connected with the eigenvalues for the adjoint action of , a fact that allows a version of the Cayley-Hamilton theorem which factorises the classical theorem (both the matrix version and the geometric algebra version).
Paper Structure (8 sections, 24 theorems, 107 equations, 1 figure)

This paper contains 8 sections, 24 theorems, 107 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The outer exponential
  3. Outer Exponential Geometry
  4. Outer trigonometry
  5. Eigenvectors and outer tangents
  6. The (outer) tangent and the Cayley transform
  7. Cayley-Hamilton for bivectors
  8. Conclusion

Key Result

Lemma 2.4

Suppose $B \in \mathbb{R}_{p,q,r}^{(2)}$ and $v \in \mathbb{R}_{p,q,r}^{(1)}$. If the image of the vector $v$ under the action of $B$ is denoted by means of $w = B \times v$, one has that for all $j$, where we define $W_0 \coloneqq 1$.

Figures (1)

  • Figure 1: The rotation of the green triangle into the orange one as a weighted sum of the untransformed (blue), point reflected (pink) and commutator (black) contributions.

Theorems & Definitions (63)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 53 more