Outer and Eigen: Tangent Concepts

TL;DR

This paper explores the use of the outer exponential of a bivector to analyze invariant decompositions and eigenvalues, leading to a novel factorization of the Cayley-Hamilton theorem in geometric algebra.

Contribution

It introduces a new perspective on invariant decomposition using the outer exponential and connects it with eigenvalues, resulting in a factorized Cayley-Hamilton theorem.

Findings

New geometric algebra approach to invariant decomposition

Connection between outer exponential and eigenvalues

Factorized Cayley-Hamilton theorem in geometric algebra

Abstract

In this paper we use the power of the outer exponential $\Lambda^B$ 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).