Reference for independence of eigenvectors of iid Gaussian matrix from its singular values.

Question

I am looking for a textbook\booklet\paper reference for the intuitive fact that if $X$ is a $n\times m$ matrix with iid Gaussian entries with zero mean, then the eigenvectors of $XX^T$ are independent of its eigenvalues. Formally: Let $X=U\Lambda V^T$ be the svd of $X$, then $U$ is independent of $\Lambda$.