Orthogonal Invariance and Identifiability

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.