  Let $A$ be a square matrix and let $\Omega$ be an open set in the plane
containing the spectrum of $A$. We consider the problem of maximizing the
operator norm $\|f(A)\|$ amongst all holomorphic functions $f$ from $\Omega$
into the closed unit disk. If $f_0$ is extremal for this problem and if
$\|f_0(A)\|>1$, then it turns out that the matrix $f_0(A)$ has special
properties, among them the fact that its principal left and right singular
vectors are mutually orthogonal. We study this class of exceptional matrices
$f_0(A)$. In particular, we are interested in the extent to which they are
characterized by the aforementioned orthogonality property.
