Eigenvalue resolution of self-adjoint matrices

TL;DR

This paper applies a resolution technique to the conjugation action of the unitary group on self-adjoint matrices, demonstrating smooth eigenvalue behavior and eigenspace decomposition on the resolved space.

Contribution

It introduces a novel application of group action resolution to analyze eigenvalues and eigenspaces of self-adjoint matrices under conjugation.

Findings

Eigenvalues are smooth on the resolved space.

The trivial bundle decomposes into smooth one-dimensional eigenspaces.

Eigenvalue resolution aids in understanding matrix spectral properties.

Abstract

Resolution of a compact group action in the sense described by Albin and Melrose is applied to the conjugation action by the unitary group on self-adjoint matrices. It is shown that the eigenvalues are smooth on the resolved space and that the trivial bundle smoothly decomposes into the direct sum of global one-dimensional eigenspaces.