Random vector whose non-invertible linear transformation is Gaussian

Question

It is well known that affine transformations of Gaussian vectors are also Gaussian.

Let $Y \sim \mathcal{N}( \mathbf{0}, I)$ be an $n$ dimensional iid Gaussian random vector. Let $A$ be an $n \times m$ matrix that does not have a left inverse. Let $A X \stackrel{d}{=} Y$. What can we say about the distribution of $X$?

Answer 1

We need $m>n$ and $A$ full rank. Then, one option is $X=A^T (AA^T)^{-1}W$, where $W\sim \mathcal N (0, I_n)$. In particular, this means that $X$ has a degenerate Gaussian distribution as it's covariance matrix $A^T (AA^T)^{-2}A^T$ is singular.