Eigenvalues of $U^* diag(\lambda_1,\ldots,\lambda_n) U$, $U$ is tall and has orthogonal columns

Question

$U\in \mathbb{C}^{n\times k}, k<n~$ 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.

Answer 1

$\min\{\lambda_1,\ldots,\lambda_n\}\|x\|^2=\min\{\lambda_1,\ldots,\lambda_n\}\|Ux\|^2\leq x^*U^*diag(\lambda_1,\ldots,\lambda_n)Ux\leq \max\{\lambda_1,\ldots,\lambda_n\}\|Ux\|^2=\max\{\lambda_1,\ldots,\lambda_n\}\|x\|^2$

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