Given real vectors $d = (d_1, \ldots, d_n)$ and $\lambda = (\lambda_1, \ldots, \lambda_n)$, where I will assume that their coefficients are arranged in non-increasing order, the Schur-Horn theorem says that there exists a Hermitian matrix with the diagonal given by $d$ and spectrum given by $\lambda$ if and only if $d \prec \lambda$, with $\prec$ denoting majorization (i.e. $ \sum_{i=1}^n d_i = \sum_{i=1}^n \lambda_i$,
and $ \sum_{i=1}^k d_i \leq \sum_{i=1}^k \lambda_i$ for all $1 \leq k < n$). Given a vector of eigenvalues, the Schur-Horn theorem then tells us all of the possible diagonal values that Hermitian matrices with such eigenvalues can take. However, it does not take into consideration the eigen*vectors* of the matrices, and the subspace which they lie in. My question is about an extension of the theorem which does.

Given a vector of eigenvalues $\lambda = (\lambda_1, \ldots, \lambda_r, 0, \ldots, 0)$ and a subspace $S$ representing the span of the eigenvectors corresponding to non-zero eigenvalues, what diagonal values can Hermitian matrices with this set of eigenvalues and with this column space take?

Does this set admit an easy characterization?