# Concentration inequality for norm of Gaussian matrices

It is known that a random matrix $A \in \mathbb{R}^{n \times n}$ whose entries are i.i.d. and follow a standard Gaussian distribution, the following upper-bound holds

$$\mathbb{E} [\| A \|] \leq \sqrt{2n \log (2n) }$$

by the Bernstein inequality. Does anyone know how the result would change if the Gaussian entries had a variance $\sigma$ and did not follow the standard Gaussian, i.e., variance = 1?

• My guess: The new random matrix become $\|\sigma A \|$, and $\mathbb{E}[\|\sigma A\|] = \sigma \mathbb{E}[\|A\|]$. Then apply the result. – BGM Aug 7 '17 at 17:40