Additive Inverse Eigenvalue Problem

• The additive inverse eigenvalue problem examines whether a square matrix A admits a nonzero diagonal matrix D such that A+D and A share the same eigenvalues, demonstrating spectral rigidity.
• The key criterion involves verifying that not all k×k principal minors of A are equal, linking combinatorial matrix properties with spectral invariance.
• Applications span numerical analysis, algebraic geometry, and quantum physics, with use cases in engineering isospectral matrices and analyzing discrete periodic operators.

The additive inverse eigenvalue problem (AIEP) asks: when does a given square matrix admit a nontrivial diagonal perturbation that preserves its spectrum? Formally, for an $n \times n$ matrix $A$ over an algebraically closed field $K$ of characteristic zero or greater than $n$, the problem is to determine whether there exists a nonzero diagonal matrix $D$ such that $\sigma(A+D) = \sigma(A)$, where $\sigma(A)$ denotes the multiset of eigenvalues of $A$. This question encapsulates both spectral rigidity under diagonal shifts and the constructive aspects of spectral assignment in matrix analysis. The problem has foundational significance across numerical analysis, algebraic geometry, and mathematical physics, with recent progress achieving a complete characterization of the matrices for which such nontrivial isospectral diagonal shifts exist (Cobb et al., 14 Jan 2026).

1. Formal Statement and Main Theorem

Given $A\in M_n(K)$, does there exist a nonzero diagonal $D = \operatorname{diag}(d_1, ..., d_n) \neq 0$ so that $A+D$ and $A$ have identical spectra? This is known as the additive inverse eigenvalue problem in rigidity form. The core result (Cobb et al., 14 Jan 2026) is:

Theorem (Cobb–Faust–Kretschmer):

Let $A\in M_n(K)$, with $K$ algebraically closed of characteristic $0$ or $>n$. There exists a nonzero diagonal $D$ with $\sigma(A+D) = \sigma(A)$ if and only if, for some $1\leq k\leq n$, not all $k\times k$ principal minors of $A$ are equal.

This criterion is both necessary and sufficient: the possibility of isospectral nontrivial diagonal perturbation is governed entirely by the structure of principal minors of the matrix.

2. Role of Principal Minors in Spectral Rigidity

AIEP is controlled by the invariance properties of the characteristic polynomial coefficients under diagonal perturbation. For any such $A$ and $D$,

$$\det(A+D-\lambda I) = \sum_{i=0}^n C_i(D)(-\lambda)^{n-i}$$

where $C_i(D)$ depends polynomially on $D$. Define spectral invariants $S_i(D) = C_i(D) - C_i(0)$; then $S_i(D)=0$ for all $i$ if and only if $\sigma(A + D) = \sigma(A)$.

Expanding $S_i(D)$ in terms of principal minors,

$$S_i(D) = \sum_{J\subseteq\{1,\dots, n\},\, 1\leq |J|\leq i} \left[ \sum_{N\subseteq [n]\backslash J,\, |N|=i-|J|} \det(A_N)\right]\prod_{j\in J}d_j$$

The leading part of $S_i$ is the elementary symmetric polynomial $e_i(d)$, whose vanishing only occurs at $d=0$ unless some set of minors differ. Therefore, if all $k \times k$ principal minors coincide for every $k$, only the trivial perturbation exists. If not, a nontrivial diagonal $D$ can be chosen with $S_i(D) = 0$ for all $i$. These facts tightly link the combinatorial geometry of submatrices to global spectral behaviour.

3. Algebro-Geometric Proof via Hilbert Schemes and the Hilbert–Chow Morphism

The proof approach synthesizes symmetric function theory, deformation of schemes, and the geometry of Hilbert schemes:

• Flat Families: The system $S_i(D) = 0$ defines a family of zero-dimensional schemes in affine space, giving a morphism $$c : \mathbb{A}^1_T \to \text{Hilb}_{n!}(\mathbb{A}^n)$$.
• Hilbert–Chow Morphism: The Hilbert–Chow morphism $\text{HC}$ sends such schemes to their support cycles in the symmetric power $\text{Sym}^{n!}(\mathbb{A}^n)$, keeping track of multiplicities and degenerations.
• Degeneration and First-Order Variation: If only $d = 0$ solved $S_i=0$, the cycle would remain rigid under degeneration, forcing all first-order variations of traces of multiplication operators to vanish.
• Extraction of Relations: Analytical extraction of the leading coefficients forces, for each $m$, that $\det(A_{[m]})$ equals the average over all $m \times m$ principal minors; iteratively, all minors must coincide. The presence of any non-coincidence yields a nontrivial solution, completing the characterization.

This algebro-geometric perspective connects the spectral rigidity of $A$ to the tangent behaviour and orbit structure under principal diagonal shifts in scheme-theoretic language (Cobb et al., 14 Jan 2026).

4. Connections and Applications: Floquet Isospectrality

A principal application occurs in the spectral theory of discrete periodic Schrödinger operators $H_V = \Delta + V$ on $\mathbb{Z}^d$, with $V$ periodic under $Q\mathbb{Z}$. Each Floquet fiber operator $L_V(z)$ has matrix size $q = \prod_i q_i$.

Two potentials $V, V'$ are Floquet isospectral if $\sigma(L_V(z)) = \sigma(L_{V'}(z))$ for all $z\in (S^1)^d$. The rigidity question for $V\equiv 0$ asks whether there exist nonzero periodic potentials $V$ isospectral to zero. By the main theorem, the only obstruction is equality of all principal minors of each $L_V(z)$. Explicitly:

• If all $q_i \leq 3$ (with at most two $q_i=3$), then no nontrivial isospectral $V$ exists;
• If some $q_j > 3$, nonzero complex periodic $V$ isospectral to zero can be constructed.

This dichotomy provides new rigidity results in the discrete periodic setting, with ramifications for solid state physics and spectral geometry, differing markedly from classical continuous isospectrality theorems of Borg and Gordon–Kappeler–Wnich (Cobb et al., 14 Jan 2026).

5. Concrete Example and Illustrative Computation

A concise illustration for $n=3$: Let

$$A = \begin{pmatrix} 0 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 2 \end{pmatrix}.$$

Its $1\times 1$ principal minors are $\{0, 1, 2\}$ (not all equal). Taking $D = \operatorname{diag}(1, -1, 0)$, one obtains $A+D = \operatorname{diag}(1, 0, 2)$ with spectrum $\{0, 1, 2\}$. Verification:

$$\det(A - \lambda I) = (0 - \lambda)(1 - \lambda)(2 - \lambda) = \det((A + D) - \lambda I),$$

thus $\sigma(A + D) = \sigma(A)$. If all $1\times 1$ minors had coincided, only $D=0$ would work.

6. Related Inverse Eigenvalue Problems and Broader Context

AIEP interacts significantly with broader inverse eigenvalue problems (IEP), where one seeks a matrix (possibly with pattern constraints) realizing a specified spectrum. In the context of graphs, the additive construction developed for block graphs allows for systematic spectrum engineering via clique appending and duplication, under the strong spectral property (SSP). In many cases, arbitrary lists of eigenvalues with controlled multiplicities can be realized, provided certain combinatorial and spectral nondegeneracy conditions are met (Lin et al., 2020). For full affine families of matrices, least squares solutions to spectral constraints (e.g., via Riemannian gradient descent/Lift-Projection methods) extend AIEP's computational paradigm (Riley et al., 10 Apr 2025).

7. Implications, Limitations, and Open Directions

The main theorem provides a definitive criterion for diagonal isospectral deformations and elucidates spectral rigidity in high-dimensional and algebraic settings. Limitations include the restriction to algebraically closed fields of characteristic $0$ or $>n$, and to square matrices (the proof relies heavily on symmetric function and scheme-theoretic formalism). A plausible implication is that similar technology may illuminate spectral rigidity for more general perturbation classes or for other structured matrix ensembles. The impact on discrete periodic operator theory and combinatorial matrix analysis highlights AIEP's central role in modern spectral theory.

References:

• Cobb, Faust, Kretschmer, "Inverse Eigenvalue Problems, Floquet Isospectrality and the Hilbert–Chow Morphism" (Cobb et al., 14 Jan 2026).
• Lin, Oblak, Šmigoc, "On the inverse eigenvalue problem for block graphs" (Lin et al., 2020).
• Kremer, Dym, and colleagues, "A Riemannian Gradient Descent Method for the Least Squares Inverse Eigenvalue Problem" (Riley et al., 10 Apr 2025).