Inverse Eigenvalue Problems, Floquet Isospectrality and the Hilbert--Chow Morphism

When can one change the diagonal of a matrix without changing its spectrum? We completely answer this question over an algebraically closed field of characteristic zero or larger than the size of the matrix: An $n \times n$ matrix $A$ admits a nonzero diagonal matrix $D$ such that $A$ and $A+D$ have the same spectrum if and only if, for some size $k$, the $k \times k$ principal minors of $A$ are not all equal. This relates to the classical additive inverse eigenvalue problem in numerical analysis and has implications for existence and rigidity results in the theory of Floquet isospectrality of discrete periodic operators in solid state physics. The proof employs new techniques involving Hilbert schemes of points and the infinitesimal structure of the Hilbert--Chow morphism.