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?