Inverse Eigenvalue Problems, Floquet Isospectrality and the Hilbert--Chow Morphism
John Cobb, Matthew Faust, Andreas Kretschmer
TL;DR
We address when adding a nonzero diagonal $D$ to an $n\times n$ matrix $A$ preserves its spectrum, proving that such a $D$ exists iff for some $k$ the $k\times k$ principal minors of $A$ are not all equal. The proof leverages spectral invariants $S_i$, the coinvariant algebra, and a flat family built from perturbations of elementary symmetric polynomials, analyzed via the Hilbert--Chow morphism on Hilbert schemes. This yields a geometric description of spectrally rigid matrices as the variety of symmetrized principal minors and connects to Floquet isospectrality for discrete periodic operators: for large periods, nontrivial isospectral complex potentials exist, while for small periods rigidity often holds. The work demonstrates a novel bridge between inverse eigenvalue problems and algebraic geometry, using Hilbert schemes to control solution behavior and multiplicities in polynomial systems arising from spectral constraints.
Abstract
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.