Let $Z \sim \mathcal{N}(0,\Sigma \otimes I)$ (so the columns of $Z$ are distributed $\mathcal{N}(0, \Sigma)$) and $A = Z'Z.$ Is there a name for the distribution on $A$? Is the density known?

I think I misunderstood the question. So is $Z$ a row vector of i.i.d. normals with mean zero and prescribed variance?
– Yemon ChoiDec 5 '11 at 6:59

$Z$ is a matrix whose columns are distributed $\mathcal{N}(0, \Sigma).$ If the rows were distributed that way, then $Z'Z$ would be a Wishart distribution.
– Alex GittensDec 5 '11 at 22:28