Graduate Student / Postdoc Seminar

Investigation of Crouzeix's Conjecture via Optimization

Speaker: Michael Overton

Location: Warren Weaver Hall 1302

Date: Friday, February 14, 2014, 1 p.m.


Crouzeix's conjecture is a fascinating open problem in matrix theory. We present a new approach to its investigation using optimization. Let \(p\) be a polynomial of any degree and let \(A\) be a square matrix of any order. Crouzeix's conjecture is the inequality

$$\|p(A)\| \leq 2 \|p\|_{W(A)}.$$

Here the left-hand side is the 2-norm of the matrix \(p(A)\), while the norm on the right-hand side is the maximum of \(|p(z)|\) over \(z\in W(A)\), the field of values (or numerical range) of \(A\). It is known that the conjecture holds if 2 is replaced by 11.08 (Crouzeix 2007).

Joint work with Anne Greenbaum, Adrian S. Lewis and Lloyd N. Trefethen