Eigenvector dynamics: theory and some applications

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