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suppose x is a gaussian vector (0 mean, known covariance $\Sigma$). Suppose A is a gaussian matrix (the kinds used in compressed sensing). Will Ax be a gaussian vector with mean 0 and a random covariance $A \Sigma A^T$? or will the distribution be something entirely different?

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By symmetry $Ax/\|Ax\|$ will be uniformly distributed on the unit sphere. – p.s. Nov 26 '11 at 19:16
up vote 4 down vote accepted

The question as you've asked it is a little bit confused. You declared that $A$ is a random matrix, and then asked if the product $Ax$ had a normal distribution with covariance matrix $A\Sigma A^T$. But that would itself be a random matrix - and that doesn't make any sense!

To make sense of your question, let's reduce it to the simplest case where $A$ is 1 x 1 and $x$ is a length 1 vector, and to keep it really straightforward suppose that the distributions of $A$ and $x$ are both standard normal.

Then the product $y=Ax$ follows the product normal distribution which is a scaled version of the modified Bessel function of the second kind.

So, no, in the more general case where $A$ is $m \times n$ and $x$ is in ${\bf R}^n$, $y=Ax$ will not be normally distributed.

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+1. Learn something new today. – Tim Nov 26 '11 at 19:31
+1 for the reduction to the simplest case. – Did Nov 26 '11 at 19:48

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