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

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

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

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

**Note.** I'm really only interested in (good) lower-bounds.

Examples
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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 $\alpha_1(A) = \operatorname{trace}(A)/n$.
- If $k=n$, then $P_V = I_n$ with probability $1$ and so $\alpha_n(A) = \lambda_\text{max}(A)$.
- If $A = I_n$, then obviously $\alpha_k(A) = 1$ for all $k \in [n]$.
- If $A = uu^\top$, a rank 1 matrix, then $\alpha_k(A) = \mathbb E_V R(A,V) = \mathbb E_V\lVert P_V u\rVert^2 = (k/n)\lVert u\rVert^2$.