A spectral function on the linear space of complex n by n matrices is an extended real valued function the that can be represented as the composition of a permutation invariant extended real valued function with the mapping that takes an n by n matrix to its spectrum counting multiplicities. The maximum modulus of the spectrum (the spectral radius) is an example of such a function. Another is the spectral abscissa which is the maximum of the real part of the spectrum. Indeed, these two examples provide much of the motivation and insight for our inquiry. Both functions are related to the stability properties of linear dynamical systems. The spectral radius has importance in the discrete case, while the spectral abscissa is important in the continuous case. Our interest in the variational properties of these functions arises from applications where one is interested in designing or ``tuning'' such systems to possess certain properties.
In this talk, we show that the tools developed in the recent book ``Variational Analysis'', by Rockafellar and Wets, are well suited to the study of these non-Lipschitzian functions. We discuss the variational calculus of such functions and apply this calculus to some specific examples.
The research presented in this talk is joint work with Michael Overton of the Courant institute NYU and Adrian Lewis of the University of Waterloo.