Eigenvectors of sums of matrices

Question

Suppose we know all eigenvalues and eigenvectors of the hermitian matrices $A$ and $B$, what does this say about the eigenvectors of $A+\varepsilon B$ for small $\varepsilon$?

Answer 1

I believe the only thing you can say is that the eigenvectors and eigenvalues will be close to those of $A$, I don't believe you can give a formula or anything. Maybe there's some way to get a first order approximation or something for the eigenvalues or eigenvectors but I'm not sure, even that seems doubtful. If the matrices commute then they are simultaneously diagonalizable so then you can give formulas.