On the geometry of the set of symmetric matrices with repeated eigenvalues

TL;DR

This paper explores the geometric structure of symmetric matrices with repeated eigenvalues, calculating their intersection volume with spheres and establishing a distance theorem, linking algebraic geometry and random matrix theory.

Contribution

It provides explicit volume calculations and a new distance theorem for symmetric matrices with repeated eigenvalues, connecting algebraic geometry and matrix theory.

Findings

Computed the volume of the set's intersection with the sphere

Proved a distance theorem analogous to Eckart-Young-Mirsky

Linked geometric properties to algebraic geometry and random matrix theory

Abstract

We investigate some geometric properties of the real algebraic variety $\Delta$ of symmetric matrices with repeated eigenvalues. We explicitly compute the volume of its intersection with the sphere and prove a Eckart-Young-Mirsky-type theorem for the distance function from a generic matrix to points in $\Delta$. We exhibit connections of our study to Real Algebraic Geometry (computing the Euclidean Distance Degree of $\Delta$) and Random Matrix Theory.