Eigenvector dynamics: theory and some applications

TL;DR

This paper develops a theoretical framework to analyze the stability of eigenvector subspaces of symmetric matrices under small perturbations, with applications in quantum physics and financial risk management.

Contribution

It introduces a general approach to study eigenvector subspace dynamics under perturbations, applicable to Gaussian orthogonal matrices and correlation matrices.

Findings

Framework effectively characterizes eigenvector stability.

Applications demonstrated with financial data.

Useful in quantum dissipation and risk control contexts.

Abstract

We propose a general framework to study the stability of the subspace spanned by $P$ consecutive eigenvectors of a generic symmetric matrix ${\bf H}_0$, when a small perturbation is added. This problem is relevant in various contexts, including quantum dissipation (${\bf H}_0$ is then the Hamiltonian) and risk control (in which case ${\bf H}_0$ is the assets return correlation matrix). We specialize our results for the case of a Gaussian Orthogonal ${\bf H}_0$, or when ${\bf H}_0$ is a correlation matrix. We illustrate the usefulness of our framework using financial data.