Let $G_{k,n}$ be the grassmannian of $k$-dimensional vector spaces of $\mathbb R^n$. By the Courant-Fisher characterization, the $k$ eigenvalue of an $n \times n$ psd matrix $A$ is given by

$$
\tag{1}
\lambda_k = \min_{V \in G_{n-k+1,n}} R(A,V),
$$
where $R(A,V):= \max_{x \in V,\,\|x\| = 1} x^\top A x$.

Now, let $V$ be drawn according to the Haar distribution on $G_{k,n}$, and replace the min in (1) by expectation over $V$.

>**Question.** What does $\alpha_k(A) := \mathbb E_V [R(A,V)]$ correspond / evaluate to ?

Examples
---

Let $P_V$ be the orthogonal projector for $V$.

- If $k=1$, then $R(A,V) = v^\top A v$, where $v$ is uniform on the unit-sphere in $\mathbb R^n$, and so $\alpha_1(A) = \mbox{trace}(A)/n$.
- If $k=n$, then $P_V = I_n$ with probability $1$ and so $\alpha_n(A) = \lambda_{\max}(A)$.
- If $A = I_n$, then obviously $\alpha_k(A) = 1$ for all $k \in [n]$.
- If $A = uu^\top$, a rank 1 matrix, then $\alpha_k(A) = \mathbb E_V R(A,V) = \mathbb E_V\|P_V u\|^2 = (k/n)\|u\|^2$.