Let $A$ be a random $m$ by $n$ rectangular sign matrix, chosen uniformly at random, with $m < n$. Let $B = A^T A$. We know, for example, that $B$ is a square and symmetric $n$ by $n$ matrix with all its diagonal entries equal to $m$ exactly. I am trying to work out how to calculate (or estimate) the expected Frobenius and spectral norm of $B$. We can assume both $m$ and $n$ are large.

How can you calculate $\mathbb{E}(||B||_F)$ and $\mathbb{E}(||B||_2)$?

The expected Frobenius norm of $B$ is defined to be

$$ \mathbb{E}(||B||_F)=\mathbb{E}\left(\sqrt{\sum_{i=1}^n\sum_{j=1}^n |b_{ij}|^2}\right) $$

where $b_{ij}$ are the elements of $B$.

The expected spectral norm of $B$ is defined to be

$$ \mathbb{E}(||B||_2)= \mathbb{E}\left(\max_{|x|_2 \ne 0}\frac{|Bx|_2}{|x|_2}\right). $$

I apologize if this turns out to be simple to do. I previously asked at https://math.stackexchange.com/questions/1517103/how-to-calculate-expected-value-of-matrix-norms-of-ata with no answer.