Part (L) of the theorem in the following paper answers your question in the case that of normal matrices.

Andreas Kriegl, Peter W. Michor, Armin Rainer: Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators. Integral Equations and Operator Theory 71,3 (2011), 407-416.
[(pdf)][1] 


  [1]: http://www.mat.univie.ac.at/~michor/DC-perturb.pdf