Consider a matrix function $A(x)$, *analytically* depending on single parameter $x$.


Consider the eigenvalue/eigenvector pair of $A(0)$, namely $\lambda(0)$ and $w(0)$.


The question is whether we can construct the analytic functions $\lambda(x)$ and $w(x)$ that are the eigenvalue/eigenvector of $A(x)$.
(I assume that both $\lambda(x)$ and $w(x)$ could have poles or branch points, hence being multi-valued).


In books I found this statement for the case of Hermitian matrix $A(t)$ only. Is it true for non-Hermitian comlex matrices as well?



Analytic perturbation of the eigenvalues/eigenvectors of non-Hermitian matrix
This is completely answered in [Kato's Perturbation theory for linear operators,](http://rads.stackoverflow.com/amzn/click/354058661X) chapter 1 (he treats the non-hermitian case). Eigenvalues are generally analytic, eigenvectors not so much (in the neighborhood of values of $A(x)$ with multiple eigenvalues.


