Consider a Hermitian matrix $H$, with each element $H_{ij} (\epsilon_1, \epsilon _2)$ being a function of two real parameters, $\epsilon_1$ and $\epsilon_2$. I am aware of the result that even if all $H_{ij}$ are analytic functions of $\epsilon_1$ and $\epsilon_2$, this does not imply that the eigenvalues of $H$ will also be analytic functions of $\epsilon_1$ and $\epsilon_2$. However, if we dilute the conditions, will the properties of the entries carry over to the properties of the eigenvalues? Specifically:
- If all matrix entries are smooth, but not necessarily analytic in the two parameters, will the eigenvalues of the matrix also be smooth?
- Suppose we dilute the conditions even further. If the entries are $C^n$ in both the variables, then can we conclude that the eigenvalues will also be $C^n$ in the variables?
- Can we make the same conclusions about eigenvectors?
Clearly, 2 would imply 1. However, I would like to know if any of these are true, or false. I would appreciate it if you could provide a proof, or refer me to resources that talk about such problems.
Thank you.