Eigenschemes and the Jordan canonical form

TL;DR

This paper explores the eigenscheme of matrices, revealing its connection to the Jordan canonical form and offering a geometric interpretation as a zero locus on projective space, providing new insights into eigenvector structures.

Contribution

It demonstrates that the eigenscheme encodes the Jordan form data and offers a geometric perspective through the tangent bundle interpretation.

Findings

Eigenscheme encodes Jordan canonical form data

Geometric interpretation as zero locus of a section

Provides an alternative viewpoint on eigenvectors

Abstract

We study the eigenscheme of a matrix which encodes information about the eigenvectors and generalized eigenvectors of a square matrix. The two main results in this paper are this decomposition encodes the numeric data of the Jordan canonical form of the matrix. We also describe how the eigenscheme can be interpreted as the zero locus of a global section of the tangent bundle on projective space. This interpretation allows one to see eigenvectors and generalized eigenvectors of matrices from an alternative viewpoint.