  The spectral abscissa is the largest real part of an eigenvalue of a matrix
and the spectral radius is the largest modulus. Both are examples of spectral
max functions---the maximum of a real-valued function over the spectrum of a
matrix. These mappings arise in the control and stabilization of dynamical
systems. In 2001, Burke and Overton characterized the regular subdifferential
of the spectral abscissa and showed that the spectral abscissa is
subdifferentially regular in the sense of Clarke when all active eigenvalues
are nonderogatory. In this paper we develop new techniques to obtain these
results for the more general class of convexly generated spectral max
functions. In particular, we extend the Burke-Overton subdifferential
regularity result to this class. These techniques allow us to obtain new
variational results for the spectral radius.
