Let $S_{n, k} = \{V \in \mathbb{R}^{n \times k} : V^T V = I_k\}$ denote the Stiefel manifold, $1 \leq k \leq n$.
Let $P \in \mathbb{R}^{n \times n}$ denote a symmetric real, positive definite matrix, and consider the matrix $$ \mathcal{M}_{k}(P) = \sqrt{P} \, \mathbb{E}\Big[V(V^T P V)^{-2} V^T\Big] \,\sqrt{P}, $$ where the expectation is taken with $V$ drawn from the uniform measure on $S_{n, k}$.
Question: What is known about $\mathcal{M}_{k}(P)$?
I do not think this can be computed in closed form, at least for general $P$ and large $n$. Thus, I am mostly interested what is known in terms of estimates (inequalities) for $\mathcal{M}_{n, k}(P)$ or functionals of this matrix (for instance, the norm, the trace, etc.)
Some (obvious) comments:
- Up to an orthogonal matrix, we can without loss of generality assume that $P$ is positive diagonal.
- We have $\mathcal{M}_k(c I_n) = \frac{k}{c n} I_n$.
