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 the column vectors (eigenvectors) with non-zero eigenvalues, but this would still be an $m \times k$ matrix where $k$ is the number of nonzero eigenvalues, so that doesn't seem to be a correct approach. Or maybe I am not applying it in a right way. Should I be taking a matrix made from "non-zero" eigenvalues and applying it to **A** instead?