  Let $A$ be a square complex matrix and $z$ a complex number. The distance,
with respect to the spectral norm, from $A$ to the set of matrices which have
$z$ as an eigenvalue is less than or equal to the distance from $z$ to the
spectrum of $A$. If these two distances are equal for a sufficiently large
finite set of numbers $z$ which are not in the spectrum of $A$, then the matrix
$A$ is normal.
