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?