Abstract
In this expository talk I will show how the field of values (or numerical range) of an operator can be used to investigate certain problems involving non-normal matrices. In particular, I will discuss decay properties and polynomial approximation of functions of large matrices, and how to estimate the rate of convergence of preconditioned iterative methods for solving the saddle point problems arising from the discretization of systems of PDEs like the Navier-Stokes equations. I will also briefly discuss Crouzeix’s conjecture, possibly the most intriguing open problem in matrix theory.
