  For any square-summable commuting family $(A_i)_{i\in I}$ of complex $n\times
n$ matrices there is a normal commuting family $(B_i)_i$ no farther from it, in
squared normalized $\ell^2$ distance, than the diameter of the numerical range
of $\sum_i A_i^* A_i$. Specializing in one direction (limiting case of the
inequality for finite $I$) this recovers a result of M. Fraas: if
$\sum_{i=1}^{\ell} A_i^* A_i$ is scalar for commuting $A_i\in M_n(\mathbb{C})$
then the $A_i$ are normal; specializing in another (singleton $I$) retrieves
the well-known fact that close-to-isometric matrices are close to isometries.
