Let $G_{n,m}$ be the Grassmanian manifold, the set of $m$-dimensional linear subspaces of $\mathbb{R}^n$.
I read in the book of Verschynin that one can sample a random element $E \in G_{n,m}$ with uniform distribution in $G_{n,m}$ by computing the column span of a random $n \times m$ Gaussian random matrix with i.i.d. $N(0,1)$ entries (meanzero with unit variance).
Can anyone give me an hint of how to prove rigorously that this object has uniform distribution in $G_{n,m})$? Where does gaussianity play a role?
The gaussianity is important because the Gaussian Orthogonal Ensemble is orthogonally invariant (which is the usual reason for Gaussian variables). The method comes from the following remark of G. W. Stewart (see the reference below):
Stewart, G. W., The efficient generation of random orthogonal matrices with an application to condition estimators. (With mircofiche section), SIAM J. Numer. Anal. 17, 403-409 (1980). ZBL0443.65027.
To generate a random orthogonal matrix uniformly, generate an i.i.d. Gaussian matrix, $G,$ compute the QR decomposition $G = QR,$ then $Q$ is the random orthogonal matrix you seek. Extension of this to arbitrary grassmannians should be easy.