Numerical Ranges and Crouzeix's Conjecture (Ryan O’Loughlin, University of Reading)

Abstract:

The numerical range of a matrix is a convex compact subset of the complex plane. This is a fundamental concept in linear algebra and finds uses in numerical analysis through approximation of eigenvalues. In this talk I will give an introduction to the numerical range of a matrix and present a new proof that the numerical range of a 2-by-2 matrix is an ellipse. If time permits, I will then give a background on Crouzeix's Conjecture, which is an open conjecture which bounds the norm of a polynomial applied to a matrix in terms of the supremum of the polynomial over the numerical range of the matrix.