  We define the discrete norm of a complex $m\times n$ matrix $A$ by
  $$ \|A\|_\Delta := \max_{0\ne\xi\in\{0,1\}^n} \frac{\|A\xi\|}{\|\xi\|}, $$
and show that
  $$ \frac c{\sqrt{\log h(A)+1}}\,\|A\| \le \|A\|_\Delta \le \|A\|, $$ where
$c>0$ is an explicitly indicated absolute constant,
$h(A)=\sqrt{\|A\|_1\|A\|_\infty}/\|A\|$, and $\|A\|_1,\|A\|_\infty$, and
$\|A\|=\|A\|_2$ are the induced operator norms of $A$.
  Similarly, for the \emph{discrete Rayleigh norm}
  $$ \|A\|_P := \max_{\substack{0\ne\xi\in\{0,1\}^m \\ 0\ne\eta\in\{0,1\}^n}}
  \frac{|\xi^tA\eta|}{\|\xi\|\|\eta\|} $$ we prove the estimate
  $$ \frac c{\log h(A)+1}\,\|A\| \le \|A\|_P \le \|A\|. $$ These estimates are
shown to be essentially best possible.
  As a consequence, we obtain another proof of the (slightly sharpened and
generalized version of the) converse to the expander mixing lemma by
Bollobas-Nikiforov and Bilu-Linial.
