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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?

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If $Z'$ means the transpose, isn't this the Wishart distribution/density? –  Yemon Choi Dec 4 '11 at 1:38
    
Oh, sorry, misread. So the columns are iid but the rows need not be? –  Yemon Choi Dec 4 '11 at 1:45
    
Isn't this related to the: en.wikipedia.org/wiki/Matrix_normal_distribution –  Suvrit Dec 4 '11 at 10:52
    
I think I misunderstood the question. So is $Z$ a row vector of i.i.d. normals with mean zero and prescribed variance? –  Yemon Choi Dec 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 Gittens Dec 5 '11 at 22:28
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