Eigenvector dynamics: general theory and some applications

TL;DR

This paper develops a general theoretical framework to analyze the stability of eigenvector subspaces under small perturbations, with applications in quantum physics and finance, providing explicit results for Gaussian orthogonal matrices and financial data.

Contribution

It introduces a novel approach using singular values of an overlap matrix to study eigenvector stability, including explicit formulas for Gaussian orthogonal matrices and covariance matrices.

Findings

Explicit spectrum of singular values for Gaussian orthogonal matrices

Application to financial covariance matrices using real data

Analysis of eigenvector angle dynamics in large eigenvalue regimes

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 financial risk control (in which case ${\bf H}_0$ is the assets return covariance matrix). We argue that the problem can be formulated in terms of the singular values of an overlap matrix, that allows one to define a "fidelity" distance. We specialize our results for the case of a Gaussian Orthogonal ${\bf H}_0$, for which the full spectrum of singular values can be explicitly computed. We also consider the case when ${\bf H}_0$ is a covariance matrix and illustrate the usefulness of our results using financial data. The special case where the top eigenvalue is much larger than all the other ones can be investigated in full detail. In particular, the dynamics of the angle made by the top eigenvector and its true direction defines an interesting new class of random processes.