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?


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