# 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.

Synopsis:

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