I have encountered a rather curious question.

Suppose I have a symmetric idempotent orthogonal projection matrix $A\in\mathbb R^{N\times N}$ that projects onto a uniformly random $n$-dimensional subspace of $\mathbb R^N$. We can generate $A$ as $VV'$ where $V\in\mathbb R^{N\times n}$ is a partial Haar orthogonal matrix. Suppose you know all the entries in $A$ except a principal minor (of size $k\times k$, say).

What can we say about the conditional distribution of $A$, i.e. its missing minor, given the observed entries in $A$?

I feel like this question must have been considered by probabilists at some point, but I haven't found a reference yet.

Since $A= VV'$ and $V$ is partial Haar orthogonal, I was thinking that this conditional distribution can be in principle written using some rather complicated integrals over constraints involving all the orthonormality conditions. Certainly that's not easy to work out, and I am not expecting a neat answer. But I do wonder if there is a simpler approach people have taken to address related questions. Similar attempts or attempts to understand conditional distributions for random projection matrices may also give some ideas.

Thanks in advance!