Let $(v_x,\lambda_x)$ be an eigenvector-eigenvalue pair of the $n\times n$ matrix $x_1A_1+\ldots+x_mA_m$. So that we have uniqueness consider some root point $x=x_0$ and fix some $(v_{x_0},\lambda_{x_0})$, and moreover assume $|v_x|=1$ for all $x$. What can we say about $\frac{d}{dx_i}v_x$ and $\frac{d}{dx_i}\lambda_x$, if we insist that $1\geq|x_i|\geq\epsilon$ for some $\epsilon$? In particular under what conditions on the matrix are the derivatives well-defined and how does it behave as we set $\epsilon\rightarrow 0$?
I'm aware this may be an involved question so I'm more interested in links to existing work than help towards a solution.