Perturbation of invariant subspaces for ill-conditioned eigensystem

TL;DR

This paper investigates the stability of invariant subspaces of diagonalizable matrices under perturbations, revealing that high eigenvector condition numbers do not always imply instability, especially near Jordan forms.

Contribution

It demonstrates that large eigenvector condition numbers do not necessarily lead to invariant subspace instability, challenging common assumptions.

Findings

High eigenvector condition number does not always cause instability.

Invariant subspaces can be stably estimated near Jordan forms.

Theoretical results apply to noisy data scenarios.

Abstract

Given a diagonalizable matrix $A$, we study the stability of its invariant subspaces when its matrix of eigenvectors is ill-conditioned. Let $\mathcal{X}_1$ be some invariant subspace of $A$ and $X_1$ be the matrix storing the right eigenvectors that spanned $\mathcal{X}_1$. It is generally believed that when the condition number $\kappa_2(X_1)$ gets large, the corresponding invariant subspace $\mathcal{X}_1$ will become unstable to perturbation. This paper proves that this is not always the case. Specifically, we show that the growth of $\kappa_2(X_1)$ alone is not enough to destroy the stability. As a direct application, our result ensures that when $A$ gets closer to a Jordan form, one may still estimate its invariant subspaces from the noisy data stably.