Lower Bound for the supreme norm of one matrix vector product Let $\Sigma$ be a covariance matrix and x be a vector. I am wondering if there is a lower bound for $||\Sigma x||_{\infty}$? I know this fact:
$$
\sigma_{min}(A)||B|| \leq ||AB||
$$
I am not sure if it can be used in my case. Thank you.

I am assuming that $x \in \mathbb{R}^n$.
$\|\Sigma x\|_\infty \ge {1 \over \sqrt{n}}\|\Sigma x\|_2 \ge {1 \over \sqrt{n}}  \sigma_\min(\Sigma) \|x\|_2 \ge {1 \over \sqrt{n}}  \sigma_\min(\Sigma) \|x\|_\infty$.
Here $\sigma_\min(\Sigma)$ is the smallest eigenvalue of $\Sigma$ (which is
positive semi definite).
