Setting: I have a $n \times n$ PSD matrix $A$ and $\tilde{A}=A+E$ be its symmetric perturbation such that $\|E\|_2=\epsilon.$ Let $(\lambda,u)$ be the principal eigenvalue, eigenvector pair of $A$ and $(\tilde{\lambda},\tilde{u})$ be that for $\tilde{A}$ (the eigenvalues are unique).
Objective: I am trying to get a tight bound for the perturbation of principal eigenvector $\|u-\tilde{u}\|_2.$
Attempts: Using the reference [Stewart and Sun 1990] used in this paper by Ng et al. (ai.stanford.edu/~ang/papers/ijcai01-linkanalysis.pdf) I can get a $O(\sqrt{n}\epsilon)$ bound (however I am unable to access the reference book). I also have a simpler proof which gives $O(\sqrt{\epsilon})$ bound.
Question: Any idea how to get a tighter (possibly $O(\epsilon)$) bound? or a condition on the error magnitude ($\epsilon \leq \delta$ ) when the overall perturbation will also be $O(\epsilon).$ Any reference note/book/papers will be helpful too.