Let $A$ be an $n\times n$ matrix of all ones. Consider the analytic perturbation of $A$ as $$\tilde{A} = A + \epsilon H_1 + \epsilon^2 H_2 + \epsilon^3 H_3 + ... $$ All matrices are symmetric. Assume $\tilde{A}$ to be positive definite. Let $E = [e_0,e_1,e_2,...e_{n-1}]$ be the eigenvectors matrix of $A$ and $\tilde{E} = [\tilde{e}_0,\tilde{e}_1,\tilde{e}_2,...\tilde{e}_{n-1}]$ be the eigenvectors matrix of the matrix $\tilde{A}$.

I have seen David Kahan theorem bounds the sine of the angle between the eigenvectors stating $\lim\limits_{\epsilon\to 0}\sin\Theta(e_i,\tilde{e}_i) = 0$ and also another theorem which says $\lim\limits_{\epsilon\to 0}\|E-\tilde{E}\|_{\infty} = 0$ or to this effect in this paper and also this paper. I am sure from these references that these statements hold when eigenvalues of $A$ are distinct.

I have seen some vague references but not sure if the above statements hold when there are eigenvalues of multiplicity greater than 1. (It is the case here, as eigenvalues of $A$ are $n$ with multiplicity $1$ and $0$ with multiplicity $n-1$.

So in this case can we say that $\lim\limits_{\epsilon\to 0} \|\tilde{E}-E'\|_{\infty} = 0$ where $E' = E$ up to a permutation of columns.

PS : I myself am not doing research in perturbation theory, but this situation arise in my work and I referred to these papers to see if the eigenvectors matrix converges. The message in those papers are a bit cryptic and was not sure if these papers apply in this case where there are eigenvalues of multiplicity higher than $1$.