Real Symmetric Matrices with Partitioned Eigenvalues

TL;DR

This paper investigates the geometric and algebraic properties of real symmetric matrices with prescribed eigenvalue multiplicities, providing formulas, parametrizations, and invariants for these matrix varieties.

Contribution

It introduces new formulas for dimension and Euclidean distance degree, and offers parametrizations and invariant descriptions for symmetric matrices with specified eigenvalue partitions.

Findings

Formulas for dimension and Euclidean distance degree.

Explicit parametrizations by rational functions.

Descriptions of invariant rings under orthogonal group action.

Abstract

We study the real algebraic variety of real symmetric matrices with eigenvalue multiplicities determined by a partition. We present formulas for the dimension and Euclidean distance degree. We give a parametrization by rational functions. For small matrices, we provide equations; for larger matrices, we explain how to use representation theory to find equations. We describe the ring of invariants under the action of the orthogonal group. For the subvariety of diagonal matrices, we give the degree.