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?
[Math] distribution of product of gaussian matrices and vectors
probabilityprobability distributions
Best Answer
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.