Suppose I'm trying to estimate the spectral radius of a square $n \times n$ matrix $A$, and let $N$ be a distribution over Gaussian i.i.d. vectors of length $n$.

Is the following lemma true: If the spectral radius of $A$ is larger than $\epsilon$ then with probability at least $1/poly(n)$, a vector $v$ sampled according to $N$ will have $\frac{|v^T A v|}{\left\|v\right\|}>\epsilon$.