Eigenvalues of symmetric and nonsymmetric
matrices are central in several areas of modern optimization,
including semidefinite programming and robust control. For symmetric
matrices, many elegant classical eigenvalue inequalities derive
from convexity properties of the spectrum; singular values enjoy
analogous inequalities and properties. I will outline the variational
theory of spectral functions, and discuss unifying algebraic frameworks,
such as semisimple Lie theory and hyperbolic polynomials. By contrast
with the symmetric case, the spectrum of a nonsymmetric matrix
is less robust and less predictive for modelling. In some contexts,
such as transient dynamics, the pseudospectrum (the set of eigenvalues
of all nearby matrices) may be a more pertinent and robust tool.
I will discuss a variety of optimization problems involving pseudospectra.
