Let $A$ be a $m\times n$ real matrix, whose entries are independent, identically distributed random variables, following standard normal distributions (mean zero and unit variance).


What is the distribution of the singular values and singular (left and right) vectors of $A$? 


For a symmetric square matrix, the concept of rotational invariance is helpful because (for ensembles where it holds) it allows us to focus on the eigenvalues since the eigenvectors are essentially uniformly distributed.


What is the analogue of "rotational invariance" in the context of a rectangular matrix like $A$? Is it helfpul here too?


Most papers/books I've read on the topic focus on square matrices so if anyone can point out relevant references I'd also appreciate that.



Singular value decomposition of random rectangular matrices
To be specific, let me assume $m\leq n$. The $m\times n$ matrix $A$ then has $n-m$ singular values equal to zero. The remaining $m$ singular values $s\_i$, $i=1,2,\ldots m$ are the positive square roots of the eigenvalues $\sigma\_i$ of the symmetric matrix product $AA^t$. The distribution $P(\sigma\_1,\sigma\_2,\ldots\sigma\_m)$ of the $\sigma\_i$'s is the [Wishart distribution.](https://en.wikipedia.org/wiki/Wishart_distribution) For $m\gg 1$ the eigenvalue density is the [Marchenko–Pastur distribution.](https://en.wikipedia.org/wiki/Marchenko%E2%80%93Pastur_distribution) 


The eigenvectors of $AA^t$ are the left eigenvectors in the singular value decomposition. These are uniformly distributed with the Haar measure in $O(m)$. The right eigenvectors are $m$ rows of length $n$ of an $n\times n$ matrix that is uniformly distributed in $O(n)$, independently of the left eigenvectors.


