Suppose $A$ is a $n \times n$ random matrix with centered Gaussian (real) i.i.d entries with variance $\sigma^2/n$.
What to we know about the spectral norm $s(A)$ of $A$, that is $\sqrt{\rho(A^t A)}$ ? (where $\rho(.)$ denotes the largest eigenvalue of a matrix)
In particular, if $A$ is symmetric, we know that $s(A)$ is precisely equal to $\rho(A)$ and from the circular law it implies that $s(A)$ converges to $\sigma$ as $n\to \infty$.
Is this last statement true for non-symmetric $A$ ?
Thanks !