Investigation of Crouzeix’s Conjecture via Nonsmooth Optimization

Michael L. Overton, Courant Institute of Mathematical Sciences, New York University

Crouzeix's conjecture is among the most intriguing developments in matrix theory in recent years.
Made in 2004 by Michel Crouzeix, it postulates that, for any polynomial p and any matrix A,
||p(A)|| <= 2 max(|p(z)|: z in W(A)), where the norm is the 2-norm and W(A) is the field
of values (numerical range) of A, that is the set of points attained by v*Av for some
vector v of unit length.  Remarkably, Crouzeix proved in 2007 that the inequality above 
holds if 2 is replaced by 11.08.  Furthermore, it is known that the conjecture holds in a 
number of special cases, including n=2.  We use nonsmooth optimization to investigate 
the conjecture numerically by attempting to minimize the “Crouzeix ratio”, defined as the
quotient with numerator the right-hand side and denominator the left-hand side of the
conjectured inequality. We present numerical results that lead to some theorems and 
further conjectures, including variational analysis of the Crouzeix ratio at conjectured global minimizers.  
All the computations strongly support the truth of Crouzeix’s conjecture.

This is joint work with Anne Greenbaum and Adrian Lewis.