Discrete norms of a matrix and the converse to the Expander Mixing Lemma

Abstract

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.