Perturbation of invariant subspaces for ill-conditioned eigensystem

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.