# Submatrix of uniform distribution on Stiefel manifold

Let $$U\in O(n,r)$$ be uniformly distributed on the Stiefel manifold. Let $$X=\begin{pmatrix} U_{11}^2 & \cdots & U_{1r}^2\\ \vdots & \ddots & \vdots\\ U_{r1}^2 & \cdots & U_{rr}^2 \end{pmatrix}$$ and $$A = (X^TX)^{1/2}$$ where the square root of a positive definite matrix is defined in the usual sense. Then, $$A$$ is a symmetric positive semidefinite random matrix distributed on $$\{M: 0\leq M\leq I\}$$.

What can we say about the distribution of $$A$$? Is there any existing result?

What is the distribution of the largest or smallest eigenvalue of $$A$$? Can we obtain the joint distribution of the eigenvalues of $$A$$?

It is trivial that when $$r=1$$, $$A$$ is just the beta distribution. What about $$r>1$$? Is it something like matrix beta distribution?