An interactive visualisation for all 2x2 real matrices, with applications to conveying the dynamics of iterative eigenvalue algorithms

TL;DR

This paper introduces two interactive visualisations for 2x2 real matrices, with v2 being applicable to all such matrices and based on Lie Sphere Geometry, aiding in understanding eigenvalue algorithms.

Contribution

The paper presents a novel interactive visualisation tool for all 2x2 real matrices, leveraging Lie Sphere Geometry to illustrate matrix properties and eigenvalue algorithm dynamics.

Findings

v2 visualisation depicts Jordan Normal Form, similarity class, and matrix properties.

v2 is intuitive despite its complexity and applicability to all 2x2 matrices.

Interactive visualisation enhances understanding of eigenvalue algorithms.

Abstract

We present two interactive visualisations of 2x2 real matrices, which we call v1 and v2. v1 is only valid for PSD matrices, and uses the spectral theorem in a trivial way -- we use it as a warm-up. By contrast, v2 is valid for *all* 2x2 real matrices, and is based on the lesser known theory of Lie Sphere Geometry. We show that the dynamics of iterative eigenvalue algorithms can be illustrated using both. v2 has the advantage that it simultaneously depicts many properties of a matrix, all of which are relevant to the study of eigenvalue algorithms. Examples of the properties of a matrix that v2 can depict are its Jordan Normal Form and orthogonal similarity class, as well as whether it is triangular, symmetric or orthogonal. Despite its richness, using v2 interactively seems rather intuitive.