Given a vector x of $n$ independent standard normal RVs, and an $m\times n$ matrix A, I was expecting that the linear transformation y=Ax would result in a multivariate normal with a covariance matrix AA$^T$. However, with $m>n$ the resulting $m \times m$ matrix AA$^T$ is not of full rank (if I understand it correctly) and hence is not invertible.

This seems to indicate that there is some "redundancy" in the system of equations y=Ax and that the same amount of information could, somehow, be represented by $n \times n$ matrix A$^*$ instead. Is this correct? And if yes, how does one construct this reduced matrix A$^*$ and then link it with the original system?

I was thinking of maybe performing a principal component analysis on AA$^T$ and then using (somehow) the column vector (eigenvectors) with non-zero eigenvalues, but this would still be an $m \times k$ matrix where $k$ is the number of nonzero eigenvalues.