Let $Q$ be a random variable taking as its values the set of $n \times k$ real matrices with orthogonal columns, and whose distribution is the Haar measure on the Stiefel manifold $O(n)/O(n-k)$. This random variable can be realized computationally by taking a random $n \times k$ matrix $A$ whose entries are drawn from i.i.d Gaussians with distribution $N(0,1)$, and then setting the corresponding value of $Q$ to be the column orthogonal matrix such that $A = QR$ in the "thin" QR factorization.

Are there any results for the asymptotic behavior of the probability distributions for the singular values of the matrix $Q(1:k,1:k)$, the upper left $k \times k$ submatrix of $Q$, as $n$ becomes large and $k \ll n$?

The application here is that any knowledge of the singular values of this truncation of these orthogonal matrices would allow us to derive probabilistic bounds on the convergence rate of certain probabilistic algorithms in matrix computations.

Edit: The best result I can find is one that computes the distribution for the largest singular value, a paper seen here. Note that I believe the authors may have made a mistake in identifying the underlying distribution as the Haar measure on the Grassmann manifold, since forming a random orthogonal matrix generates an ordered rather than unordered orthonormal basis.


On a closer look at the paper mentioned above, it turns out that the joint pdf of the singular values $\lambda_1, \ldots, \lambda_k$ is actually classically known from Muirhead's Aspects of Multivariate Statistical Theory and is reproduced in equation (1) of the paper.

Given how nasty these sorts of probability distributions are, I doubt that any else can be done to massage these expressions into a nicer form.


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