Singular vector of random Gaussian matrix Suppose $\Omega$ is a Gaussian matrix with entries distributed i.i.d. according to normal distribution $\mathcal{N}(0,1)$. Let $U \Sigma V^{\mathsf T}$ be its singular value decomposition. What would be the distribution of the column (or row) vectors of $U$ and $V$? Would it be a Gaussian or anything closely related?

The original question suggested IID entries, which means that in the limit that the matrix gets big, the singular values follow a Marchenko-Pastur distribution.
