I asked this (with background) here https://stats.stackexchange.com/questions/38494/principal-component-analysis-bootstrap-and-probability-of-eigenvalue-collision

but did not really get any answers. See that post for the background.

Let $D$ be some open set in the plane, say. Not really important where the set $D$ sits, but it shoud not be only a line/curve. Suppose we have defined a continuous function on $D$ $$ f \colon D \mapsto \text{Sym}^n $$ where $\text{sym}^n$ is the set of (real) symmetric $n \times n$ matrices. How can I define the eigenvectors of $f(x), x \in D$ as a continuous function on $D$? How can I calculate this? And how can I deal with eigenvalue collisions? A simple example clarifying this point (and defined on a curve): Let $$ f(t) =\left( \begin{matrix} 1+t & 0 \cr 0 & 1-t \end{matrix}\right) $$ Then the largest eigenvalue is $$ \lambda_1(t) = 1+ |t| $$ but the eigenvector corresponding to the largest eigenvalue cannot be defined as a continuous function: $$ v_1(t) = \begin{cases} e_2 & t\le 0 \cr e_1 & t > 0 \end{cases} $$ So what I want is to look at the two eigenvalue functions $1+t, 1-t$ and follow the eigenvectors corresponding to each one, which obviously can be done in a continuos (constant!) manner.

ADDED after the answer by Anthony Quas:

Is it possible to give some further conditions, under which a solution is possible? Differentiability? Or, if the matrices are realizations of some random field of matrices, can something be said about the probability some continuous selection is possible?