Lucky Hans wrote:
Let P be a random orthogonal projection on R^d, chosen according to the "uniform distribution" on the set of rank k orthogonal projections. Are the squares of any two distinct entries on the diagonal uncorrelated?
(It probably doesn't matter but I assume that k is at least 2 and d at least 2k.)
Let's do a simple calculation with d=4 and k=2, which has a reasonably nice coordinate system to work with.
The Haar measure on SO(4) is of course induced by
. So upon identifying
, we might as well take 1,i as the two orthonormal vectors, and have
are random unit quaternions and
is the projection to
. Now, writing
, we get
Switching to (spherical) polars, (and for brevity
So now take expectations of
, and remember the (normalised) measure on
is given by
and you should see the covariance is nonzero. (You can cheat using Weyl's integration formula instead of passing to the explicit description of Spin(4).)