Eigenvalues of $U^* diag(\lambda_1,\ldots,\lambda_n) U$, $U$ is tall and has orthogonal columns $U\in \mathbb{C}^{n\times k}, k<n~$ and
$~\lambda_i\in \mathbb{R} ~\forall~ i$
$U^*U=I_k$ but $UU^*$ is unknown.
Note that, $U$ is tall matrix formed by a few columns of some unitary matrix.
This matrix-form seems to be similar to an Eigen-decomposition,
but I fail to see any relation between the eigenvalues of  $~U^* diag(\lambda_1,\ldots,\lambda_n) U~$ and $(\lambda_1,\lambda_2,\ldots)$
Another observation (if it helps in anyway):
$U^*diag(\lambda_1,\ldots,\lambda_n)U$ also appears like a $k\times k$ sub-matrix of another $n\times n$ matrix, unitarily similar to $diag(\lambda_1,\ldots,\lambda_n)$.
In the worst case, I would want at least the maximum eigenvalue & eigenvector of it.

$\min\{\lambda_1,\ldots,\lambda_n\}\|x\|^2=\min\{\lambda_1,\ldots,\lambda_n\}\|Ux\|^2\leq x^*U^*diag(\lambda_1,\ldots,\lambda_n)Ux\leq \max\{\lambda_1,\ldots,\lambda_n\}\|Ux\|^2=\max\{\lambda_1,\ldots,\lambda_n\}\|x\|^2$ 
and therefore $\min\{\lambda_1,\ldots,\lambda_n\}\leq \lambda_i(S)\leq \max\{\lambda_1,\ldots,\lambda_n\}$ for all the eigenvalues of $S:=U^*diag(\lambda_1,\ldots,\lambda_n)U$.
