  A unitarily invariant projective framework is introduced to analyze the
complexity of path-following methods for the eigenvalue problem. A condition
number, and its relation to the distance to ill-posedness, is given. A Newton
map appropriate for this context is defined, and a version of Smale's
$\gamma$-Theorem is proven. The main result of this paper bounds the complexity
of path-following methods in terms of the length of the path in the condition
metric.
