I was trying see if there was a way to prove the property of linear transformations of gaussian distributions without using the characteristic function.
I came across the answer at the bottom:
Proof of the affine property of normal distribution for a landscape matrix
Where the writer references the generalized inverse. This appears to work except I get stuck at the final covariance matrix formula as I don't see an obvious way to collapse the generalized inverse into the covariance matrix. Using the pseudo-inverse it comes down to showing that:
$$ ((A^TA)^{-1}A^T)^T \Sigma^{-1} ((A^TA)^{-1}A^T) = (A \Sigma A^T)^{-1} $$
I suspect this can be done but I'm not sure how.