# Parametric distributions over orthogonal projection matrices?

Consider the set of rank $m$ orthogonal projection matrices in $\mathbb{R}^{d\times d}$, for some $m<d$. With appropriate measure-theoretic considerations, one can define multiple distributions on this set, e.g., the uniform distribution, or the marginal distribution of the projection onto the subspace spanned by $m$ random vectors (say, i.i.d. standard normal, which results in the uniform distribution).

Are there any parametric distribution families of this kind which have been studied previously? I'm specifically interested in the typical properties, like the mean, variance, higher moments, marginal distributions, etc.

For example, has the distribution of the projection onto the span of $m$ iid $\mathcal{N}(0,\Sigma)$ vectors been studied? I know the uniform case is used extensively in compressed sensing, but I don't recall seeing the non-uniform case anywhere.

• Do the rank of the element and the quantity of random vectors are related by the same number $m$? May 30, 2015 at 2:23
• Well, in that example, sure -- $m$ iid normal vectors would almost surely be linearly independent, and so the projection onto their span would be rank $m$. May 30, 2015 at 6:10
• I would not be sure there is something already digested for multidimensional distributions. After the normal and wishart there is nothing much more highlighted on the standard cases.... And the multivariate normal distribution as projections is indeed extensively used through PCA, Hotelling Tests and flavours like that.. From the theoretical way, surely it has been formulated as algebras over varieties, but i dont have any particular reference for that... May 30, 2015 at 6:49
• I would seek too onto Bayesian Analysis. Probably there is good references there for the higher dimensional case, and under this analysis it is habitual to perform conditional distributions -which you didnt mention, but which are somehow related to 'projections'- May 30, 2015 at 6:51
• The iid Normal case is the uniform distribution (on the Grassmann manifold).
– whuber
Jan 2, 2017 at 10:46