Inverse Symplectic Eigenvalue Problem

• The inverse symplectic eigenvalue problem is a class of inverse problems that determine if a matrix with a prescribed symplectic spectrum and structure exists.
• It integrates graph-constrained sparsity patterns and structure-preserving eigenvalue modifications, employing tools like the strong symplectic spectral property.
• The problem has key applications in quantum mechanics, control theory, and numerical linear algebra by linking spectral properties with geometric and combinatorial constraints.

The inverse symplectic eigenvalue problem encompasses a family of inverse problems central to symplectic geometry, matrix analysis, and spectral graph theory. Its aim is to determine, for a prescribed symplectic spectrum or partial spectrum, whether there exists a matrix—subject to symplectic, symmetry, or structural constraints—realizing that spectrum. Modern variants include prescribed sparsity patterns induced by graphs, structure-preserving eigenvalue assignment for symplectic matrices and pencils, and realization in specialized matrix classes such as normal $J$-symplectic or (skew) $J$-Hamiltonian matrices. Insights into this problem have substantial impact in mathematical physics, control, combinatorics, and numerical linear algebra.

1. Symplectic Eigenvalues and Williamson's Theorem

A $2p \times 2p$ real symmetric positive definite matrix $A$ possesses %%%%4%%%% positive symplectic eigenvalues $0 < d_1 \leq \cdots \leq d_p$, uniquely defined up to ordering, arising from Williamson's theorem. Specifically, there exists a symplectic matrix $S$ such that

$$A = S^T \begin{pmatrix} D & 0 \ 0 & D \end{pmatrix} S, \quad D = \operatorname{diag}(d_1, \ldots, d_p)$$

where $S^T \Omega_{2p} S = \Omega_{2p}$ for the standard symplectic form $$\Omega_{2p} = \begin{pmatrix} 0 & I_p \ -I_p & 0 \end{pmatrix}$$. The symplectic spectrum, denoted $\operatorname{SympSpec}(A)$, coincides with the moduli of eigenvalues of the Hamiltonian matrix $\Omega_{2p}A$, that is, $$\operatorname{spec}(\Omega_{2p}A) = \{ \pm i\,d_1, ..., \pm i\,d_p \}$$ (Gupta et al., 17 Jan 2026).

2. Inverse Symplectic Eigenvalue Problem on Graphs

Given a labeled simple graph $G^L$ on $n = 2p$ vertices, let $$\mathcal{S}(G^L) = \{A \succ 0 : A_{ij} \neq 0 \iff i = j \text{ or } ij \in E(G^L)\}$$ denote the family of real symmetric positive definite matrices with prescribed sparsity. The inverse symplectic eigenvalue problem (ISEP-$G$) asks: for a prescribed multiset $\{d_1, ..., d_p\} \subset (0, \infty)$, does there exist $A \in \mathcal{S}(G^L)$ with $\operatorname{SympSpec}(A) = \{d_1, ..., d_p\}$? In addition, the inverse symplectic multiplicity problem seeks all possible multiplicity patterns, and the maximum symplectic multiplicity problem asks for bounds on the largest possible multiplicity for a fixed symplectic eigenvalue. These variants connect the classical inverse eigenvalue problem to graph-theoretic structures and symplectic geometry (Gupta et al., 17 Jan 2026).

Theoretical advances include the "strong symplectic spectral property" (SSSP), which characterizes when intrinsic matrix perturbations (generated by the symplectic Lie algebra) combined with the coordinate space spanned by $G^L$ suffice to reach all symmetric directions in $\operatorname{Symm}(2p)$ (Gupta et al., 17 Jan 2026).

3. Structure-Preserving Eigenvalue Modification for Symplectic Matrices

Structure-preserving eigenvalue updates for symplectic matrices generalize classical results such as Brauer's and Rado's theorems to symplectic settings. If $S \in \mathbb{R}^{2n \times 2n}$ is symplectic and $x_1, x_2$ are eigenvectors for eigenvalues $\lambda_1, \lambda_1^{-1}$, one seeks a minimal-rank update so that the spectrum of the updated matrix replaces $\lambda_1, \lambda_1^{-1}$ by any prescribed pair $\mu, \mu^{-1}$, preserving symplecticity and leaving all other eigenvalues unchanged.

The explicit update is

$S' = S + X R X^T J^T S$

where $$X = [x_1 \; x_2],\; X^T J X = J_2,\; R \in \mathbb{C}^{2 \times 2}$$ solves two coupled matrix equations enforcing trace/determinant and symmetry restrictions. Specific forms for $R$ yield minimal distance in norm between $S$ and $S'$. This approach provides universal norm bounds on the perturbation, explicit control over Segre characteristics, and guarantees on eigenvalue condition numbers (Saltenberger, 2023).

These results extend to symplectic matrix pencils $P(\lambda) = A - \lambda B$, with explicit formulae for structure-preserving eigenvalue modification of the pencil (Saltenberger, 2023).

4. Normal $J$-Symplectic and (Skew) $J$-Hamiltonian Variants

Let $J \in \mathbb{R}^{2n \times 2n}$ be a real normal matrix with $J^2 = -I$, $J^T = -J$. A matrix $A \in \mathbb{C}^{2n \times 2n}$ is $J$-symplectic if $A J A = J$, and is normal if $A A^* = A^* A$. Normal $J$-symplectic matrices satisfy a fourfold symmetry of the spectrum: $$\lambda \in \sigma(A) \implies \left\{ \lambda, \overline{\lambda}, \frac{1}{\lambda}, \frac{1}{\overline{\lambda}} \right\} \subset \sigma(A)$$ The inverse eigenvalue problem asks: for a prescribed spectrum (closed under this symmetry) and a data matrix $X$ of eigenvectors, does there exist a normal $J$-symplectic $A$ with $A X = X D$? The problem admits a reduction via the Cayley transform to the normal $J$-Hamiltonian case, leading to necessary and sufficient feasibility conditions formulated as block matrix equations (Gigola et al., 2024).

An explicit algorithm is provided: block-diagonalize $J$, partition $X$, solve matrix equations for the auxiliary variable, apply the Cayley transform, and recover $A$. Procrustes minimization may be applied to find a solution closest in Frobenius norm to a prescribed initial guess (Gigola et al., 2024).

5. Tools and Results for Graph-Constrained ISEP

Significant structural and algorithmic tools have been developed for ISEP-$G$:

• Strong Symplectic Spectral Property (SSSP): Characterizes when a graph supports full spectrum assignment and allows propagation of realizable spectra to supergraphs via the Supergraph Theorem.
• Bifurcation Theorem: For $N$ with SSSP, any sufficiently small perturbation of the symplectic spectrum can be achieved within the same sparsity pattern.
• Matrix Liberation Lemma: Allows addition of nonzero entries in $N$ while preserving the symplectic spectrum, using tangent directions generated by the symplectic Lie algebra.
• Couplings and Coupled Zero-Forcing: Encodes labelings compatible with symplectic blocks and provides combinatorial upper bounds on the possible multiplicities of any symplectic eigenvalue. The coupled zero-forcing number $Z_C(G^C)$ gives the maximal possible eigenvalue multiplicity for any symplectic eigenvalue realizable on $G^C$ (Gupta et al., 17 Jan 2026).

6. Extremal, Classification, and Algorithmic Results

For symplectic positive definite $N \in \mathcal{S}(G^L)$ with all symplectic eigenvalues equal to one, $N$ itself must be symplectic. In this case, sharp lower bounds for the number of nonzero entries are obtained: for connected $G^L$,

$\#\{i \leq j : N_{ij} \neq 0\} \geq 4n - 4$

achieved for the "triangular path" $TP_{2p}$, a chain of $p-1$ triangles plus a pendant. For any irreducible positive definite matrix $M$, the sum of nonzero entries in $M$ and $M^{-1}$ obeys

$$\#\{M_{ij} \neq 0\} + \#\{(M^{-1})_{ij} \neq 0\} \geq 8n - 8$$

(Gupta et al., 17 Jan 2026).

Symplectic spectrally arbitrary graphs—where every positive multiset can be realized as the symplectic spectrum—include all complete graphs, joins of two cliques, bicliques, and path-of-triangles. Classification for graphs of order four is complete: for each unlabeled graph and coupling, either every multiset can be realized, or only multisets with simple spectrum occur. For paths and trees such as $P_4$ and $K_{1,3}$, only spectra with distinct entries appear (Gupta et al., 17 Jan 2026).

Algorithmic procedures are constructed for extending a solution to supergraphs (via incremental edge addition and verification of SSSP) and explicit construction of symplectic positive definite matrices with prescribed sparsity or extremal properties.

7. Connections, Applications, and Broader Context

The inverse symplectic eigenvalue problem arises in quantum mechanics, control theory, and signal processing, especially in contexts where preservation of the symplectic structure encodes fundamental physical or geometric constraints. Its graph-constrained variants link matrix analysis with combinatorics and zero-forcing techniques, relating spectral multiplicities to structural graph invariants. Structure-preserving modifications to spectra are relevant in robust control, model reduction, and Hamiltonian dynamical systems.

A plausible implication is that further generalizations to other matrix classes with symmetry or sparsity constraints, combined with algorithmic advances for feasibility testing and extremal realization, will continue to drive the field, especially with applications emerging from symplectic geometry-driven data analysis and graph-theoretic models in physical and mathematical systems (Saltenberger, 2023, Gupta et al., 17 Jan 2026, Gigola et al., 2024).