Eigenvectors of sum of Hermitian matrices

Question

Given two real Hermitian matrices $A$ and $B$, what can one say about the eigenvectors of $A+ \epsilon B$ in relation to $A$? Here $\epsilon \in [0,1]$ and $\epsilon B$ is a slight perturbation.

Answer 1

The following is a result of Bhatia's Matrix Analysis:

Theorem VII.3.2: (simplified) Let $A,B$ be Hermitian operators, and let $S_1,S_2$ be subsets of $\Bbb R$ such that $\operatorname{dist}(S_1,S_2) = \delta > 0$. Let $E = P_A(S_1), F = P_B(S_2)$. Then if $\|\cdot\|$ denotes the spectral norm (largest singular value), then we have $$ \|E - F^\perp\| = \|EF\| \leq \frac{\pi}{2\delta}\|A - B\| $$

Here, $P_A(S)$ denotes the projection onto the eigenspace of $A$ associated with the elements of $S$, and $\operatorname{dist}(S_1,S_2) = \inf_{x \in S_1, y \in S_2} |x - y|$. Perhaps this will lead to something useful in your case.