Perturbation of invariant subspaces for ill-conditioned eigensystem

02/28/2022

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by   He Lyu, et al.

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Given a diagonalizable matrix A, we study the stability of its invariant subspaces when its matrix of eigenvectors is ill-conditioned. Let 𝒳_1 be some invariant subspace of A and X_1 be the matrix storing the right eigenvectors that spanned 𝒳_1. It is generally believed that when the condition number κ_2(X_1) gets large, the corresponding invariant subspace 𝒳_1 will become unstable to perturbation. This paper proves that this is not always the case. Specifically, we show that the growth of κ_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.