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

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