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I am reading this paper:

http://society.math.ntu.edu.tw/~journal/tjm/V16N1/TJM-258.pdf

where the authors find additive perturbation bounds on the matrix of the eigenvectors of a Hermitian matrix. I am not a mathematician, and I am struggling to understand their Theorem 2.1, where actually they propose this bound.

My understanding, is that it can be simplified as follows: Let $A\in \mathbb{C}^{n\times n}$ be an Hermitian matrix, with eigendecomposition $$ A = U \Lambda U^{'},\,\,\,\,\,\Lambda = diag(\lambda_1,...,\lambda_n) $$ and $$ \tilde{A} = A + \Delta_A,\,\,\,|| \Delta_A ||_F =\epsilon $$ a perturbed, Hermitian matrix, with eigendecomposition $$ \tilde{A} = \tilde{U} \tilde{\Lambda} \tilde{U}^{'} $$ Then, $$ ||U - \tilde{U}||_F = \frac{\epsilon}{\min_{i\neq j}|\lambda_i-\lambda_j|} + O(\epsilon^2),\,\,\,\,\,\,\,\,\,\epsilon\to 0 $$

Does this make any sense? There exist additive perturbation bounds on the eigenvectors of a Hermitian matrix?

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  • $\begingroup$ This should follow from the standard perturbation theory for the non-degenerate case: en.wikipedia.org/wiki/… (see especially the formula for $|n^{(1)}\rangle$ near the bottom of the section; the article is written in a physics style, but it's easy to make these things rigorous) $\endgroup$ Mar 8, 2017 at 0:44

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