TITLE: Conditions for smooth dependence of the eigenvalues and eigenvectors of a matrix on a set of parameters
QUESTION [19 upvotes]: Let $A\in\mathcal M_n$ be an $n\times n$ real [symmetric] matrix which depends smoothly on a [finite] set of parameters, $A=A(\xi_1,\ldots,\xi_k)$. We can view it as a smooth function $A:\mathbb R^k\to\mathcal M_n$.

1. What conditions should the matrix $A$ satisfy so that its eigenvalues
  $\lambda_i(\xi_1,\ldots,\xi_k)$,
  $i=1,\ldots,n$, depend smoothly on the
  parameters $\xi_1,\ldots,\xi_k$?

e.g. if the characteristic equation is $\lambda^3-\xi=0$, then the solution $\lambda_1=\sqrt[3] \xi$ is not derivable at $\xi=0$.

2. What additional conditions should the matrix $A$ satisfy so that we can
  choose a set of eigenvectors
  $v_i(\xi_1,\ldots,\xi_k)$,
  $i=1,\ldots,n$, which depend smoothly
  on the parameters
  $\xi_1,\ldots,\xi_k$?

Update - important details

The domain is simply connected
The rank of $A$ can change in the domain
The multiplicities of the eigenvalues can change in the domain, they can cross
The matrix $A$ is real symmetric
$n$ and $k$ are finite

Update 2

A relaxation of the conditions of the problem:
For fixed $p=(\xi_{01},\ldots,\xi_{0k})$, can we find an open neighborhood of $p$ in the domain and a set of conditions ensuring the smoothness of the eigenvalues and the eigenvectors?

REPLY [4 votes]: See the following papers by Armin Rainer (et al):
Perturbation theory for normal operators. arXiv:1111.4475
Quasianalytic multiparameter perturbation of polynomials and normal matrices.
arXiv:0905.0837.
Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators. Integral Equations and Operator Theory 71,3 (2011), 407-416.
arXiv:0910.0155