Investigation of Crouzeix's Conjecture via Nonsmooth Optimization
Michael Overton
Courant Institute of Mathematical Sciences
Abstract:
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)|| <= 2max(|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 righthand 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.