Let $d > n$, and suppose that $G \in \mathbb{R}^{n \times d}$ is a random matrix with iid standard Normal entries. Let $K_G$ denote the (random) kernel of this matrix. It has codimension $n$ with probability $1$. 

Consider the following ``eigenvalue within a subspace'' for the diagonal operator corresponding to the positive sequence $a = (a_1, \dots, a_d)$, 
$$
\lambda_a(G) = 
\sup_{u \in K_G} \Big\{\, \sum_{j=1}^d u_j^2 : \sum_{j=1}^d \frac{u_j^2}{a_j} \leq 1\,\Big\}
$$

What is $\mathbb{E}[\lambda_a(G)]$? 

Two obvious observations: 
- In the special case that $a_1 = a_2 = \cdots= a_d \equiv \alpha$, then evidently
$$
\lambda_a(G) = \alpha, 
$$
with probability $1$, and the supremum is attained by taking any unit vector in $K_G$ and scaling it appropriately.
- We can define $\tilde \lambda_a(K) = \sup_{u \in K}  \{ \sum_{j=1}^d u_j^2 : \sum_{j=1}^d u_j^2/a_j \leq 1\}$, where $K \subset \mathbb{R}^d$ is a subspace of codimension $n$. Then $\mathbb{E}[\lambda_a(G)] = \mathbb{E}[\tilde \lambda_a(K)]$ where the latter expectation is with respect to Haar measure over $K$ with codimension $n$.