  Orthogonally invariant functions of symmetric matrices often inherit
properties from their diagonal restrictions: von Neumann's theorem on matrix
norms is an early example. We discuss the example of "identifiability", a
common property of nonsmooth functions associated with the existence of a
smooth manifold of approximate critical points. Identifiability (or its
synonym, "partial smoothness") is the key idea underlying active set methods in
optimization. Polyhedral functions, in particular, are always partly smooth,
and hence so are many standard examples from eigenvalue optimization.
