Imaginary eigenvalues and complex eigenvectors explained by real geometry

TL;DR

This paper explains imaginary eigenvalues and complex eigenvectors using real geometric algebra, providing geometric interpretations in two, three, and four dimensions, and connecting algebraic properties to rotations.

Contribution

It introduces a real geometric algebra approach to interpret eigenvalues and eigenvectors, extending the understanding from 2D to higher dimensions.

Findings

Imaginary eigenvalues correspond to rotations in real geometric algebra.

Eigenvectors are two-component eigenspinors reducible to vector duplets.

The approach generalizes to three and four dimensions.

Abstract

This paper first reviews how anti-symmetric matrices in two dimensions yield imaginary eigenvalues and complex eigenvectors. It is shown how this carries on to rotations by means of the Cayley transformation. Then a real geometric interpretation is given to the eigenvalues and eigenvectors by means of real geometric algebra. The eigenvectors are seen to be \textit{two component eigenspinors} which can be further reduced to underlying vector duplets. The eigenvalues are interpreted as rotation operators, which rotate the underlying vector duplets. The second part of this paper extends and generalizes the treatment to three dimensions. Finally the four-dimensional problem is stated.