Variational Analysis of the Crouzeix Ratio

Let $W(A)$ denote the field of values (numerical range) of a matrix $A$. For any polynomial $p$ and matrix $A$, define the Crouzeix ratio to have numerator $\max\left\{|p(\zeta)|:\zeta\in W(A)\right\}$ and denominator $\|p(A)\|_2$. M.~Crouzeix's 2004 conjecture postulates that the globally minimal value of the Crouzeix ratio is $1/2$, over all polynomials $p$ of any degree and matrices $A$ of any order. We derive the subdifferential of this ratio at pairs $(p,A)$ for which the largest singular value of $p(A)$ is simple. In particular, we show that at certain candidate minimizers $(p,A)$, the Crouzeix ratio is (Clarke) regular and satisfies a first-order nonsmooth optimality condition, and hence that its directional derivative is nonnegative there in every direction in polynomial-matrix space. We also show that pairs $(p,A)$ exist at which the Crouzeix ratio is not regular.

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Submitted to Math Programming