I've been searching without much luck for a reference that can help with the following problem:
Let $R$, $S$, and $E$ be symmetric matrices, with $R = S + E$, where $E$ is a small perturbation. Then let $V_S$ and $V_E$ be unitary matrices diagonalizing $S$ and $E$. Can we pick a matrix $V_R$ which diagonalizes $R$ in order to get an upper bound for $||V_R - V_S||$ in terms of $||V_S - V_E||$?
Observe that in the extreme case where $V_S = V_E$, then $S$ and $E$ are simultaneously diagonalizable and we can pick $V_R = V_S$. This leads me to believe that if $V_S$ and $V_E$ are "close" we might be able to make a statement about $V_R$ and $V_S$ being "close."
Most results I've found have focused on bounding $||V_R - V_S||$ in terms of $||E||$; see Lemma 4.3 in this paper. Although the authors just show that $||V_R - V_S||$ is $O(||E||)$, I've been able to upper bound by a specific constant using the Davis-Kahan $\sin(\theta)$ theorem. However, I'm looking for a bound in terms of the eigenbasis misalignment $||V_S - V_E||$. Kato's Perturbation Theory for Linear Operators chapter 2 seemed promising, but it mainly deals with parameterizations of pertubations and doesn't look directly applicable.
Kind of at a loss for how to make progress here, any related results / references would be much appreciated!