Suppose I look at the set of matrices which are invertible and satisfy $$ \left\|A-Id\right\|_{op}<r $$ for some $r<1$, where $Id$ is the $n\times n$ identity matrix. An $\epsilon$-net of such a set would consist of finitely many matrices which are each at least $\epsilon$ apart from each other in the operator norm, and satisfying that every other matrix in this ball is at least $\epsilon$ close to one of these.
My question is this: what is the minimal cardinality of such a net? It would be interesting to me if it were something strictly less than $\left(C/\epsilon\right)^n$.
