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](http://mathworld.wolfram.com/FrobeniusNorm.html) and [spectral norm](http://mathworld.wolfram.com/SpectralNorm.html) 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.



How to calculate expected value of matrix norms of $A^TA$?
Expanding a bit on Yemon Choi's comment: concentration is indeed the key. First,
$\|B\|\_F^2$ is simply the sum of the squares of singular values of $A$, and 
$E\|B\|\_F^2=m(m-1)n+mn^2$. On the other hand by standard concentration inequalities (using that $\|B\|\_F^2=\sum \sigma\_i^4$ where $\sigma\_i$ are the singular values of $A$), you obtain an exponential (in fractional power of m) concentration around the mean, which is good enough to also conclude that 
$E\|B\|\_F/\sqrt{m^2n+mn^2}\to 1$ (as both $m$ and $n$ grow)..


For the spectral norm, one can argue similarly: the top singular value also concentrates around its mean, and converges to the edge of the spectrum of the pastur-marchenko law. This is explained in great detail in Bai and Silverstein's book on spectral analysis of random matrices. 


