Given a $n \times n$ invertible matrix $A$, I am interested in the set

$$ \mathcal{S}(A) = \{ D \textrm{ diagonal matrix } \mid \det(D - A) = 0 \}. $$

Thus, for all eigenvalues $\lambda_i$, we have $\lambda_i I \in \mathcal{S}(A)$, where $I$ is the identity matrix.

I would like to learn more about this set $\mathcal{S}(A)$. Is there a name for it in literature? In particular, I am interested whether it can be parametrized in terms of the eigenvalues and eigenvectors of $A$. Further, I am interested in unitary $U$ and $ \mathcal{S}_\textrm{1} \subset \mathcal{S}(U)$ such that for $D \in \mathcal{S}_\textrm{1}$ that $$|d_{11}| = \dots = |d_{nn}| = 1$$