Strong Symplectic Spectral Property (SSSP)

• SSSP is a central symplectic linear algebra property that establishes explicit spanning criteria ensuring canonical forms and the solvability of inverse symplectic eigenvalue problems.
• It facilitates the derivation of block-diagonal Darboux and polarization forms for self-adjoint operators, extending classical spectral theorem techniques.
• Algorithmic methods such as rank verification and Hamiltonian perturbations leverage SSSP to construct sparse positive definite matrices with prescribed symplectic spectra.

The Strong Symplectic Spectral Property (SSSP) is a central property in symplectic linear algebra, relevant to both the structure theory of self-adjoint operators on symplectic vector spaces and the inverse symplectic eigenvalue problem for positive definite matrices in real or complex settings. SSSP establishes explicit structural and spanning criteria that guarantee the existence of canonical forms and the ability to solve spectrum realization problems with robust constraints. Its first rigorous formulation appeared in the context of self-adjoint operators over perfect fields (characteristic not 2), where it guarantees a block-diagonal Darboux normal form and a decomposition via polarization and Lagrangian subspaces (Malagón, 2017). More recently, SSSP has become a fundamental technical tool for solving the inverse symplectic eigenvalue problem on graphs, enabling constructive realization theorems and characterizations of positivity, sparsity, and spectral multiplicity (Gupta et al., 17 Jan 2026).

1. Formal Definition and Algebraic Characterizations

For $n=2p$ even, let $\Omega=\begin{pmatrix}0&I_p\-I_p&0\end{pmatrix}$ be the standard symplectic matrix, and $Sp(n)$ the real symplectic group. For a real symmetric $n\times n$ positive definite matrix $N\succ0$, SSSP concerns the following linear-algebraic spanning property:

$$\left\{ N M + M N : M \in \mathfrak{sp}(n)\right\} + \mathrm{span}\,S(G^L) = \mathrm{Symm}(n)$$

where $\mathfrak{sp}(n)$ is the Lie algebra of Hamiltonian matrices (all $n\times n$ real matrices of the form $\begin{pmatrix}R&E\F&-R^\top\end{pmatrix}$ with $E,F$ symmetric), and $S(G^L)$ is the linear span of coordinate matrices dictated by the edge labeling of a graph $G^L$. Equivalently, as shown in [(Gupta et al., 17 Jan 2026), Theorem 5.4], $N$ has SSSP if and only if the only symmetric $Y$ satisfying

$$N\circ Y=0 \qquad \text{and} \qquad \Omega N Y = Y N \Omega$$

is $Y=0$, with $N\circ Y$ denoting the Hadamard (entrywise) product. This characterization connects Hadamard sparsity to Hamiltonian equivariance, producing a nondegeneracy condition fundamental to spectral synthesis and inverse problems.

2. SSSP for Self-Adjoint Operators on Symplectic Spaces

For a $2n$-dimensional symplectic vector space $(V,\omega)$ over a perfect field of characteristic not 2, and an $\omega$-self-adjoint endomorphism $f:V\to V$ ($f^*=f$), the SSSP guarantees the following canonical forms (Malagón, 2017):

1. Polarization Form: There exists a Lagrangian subspace $U\subset V$ and a linear map $l:U\to U$ such that under the explicit symplectic isomorphism $\Phi:U\oplus U^*\to V$, $f = \Phi \circ (l, l^*) \circ \Phi^{-1}$, with $l^*$ the dual operator.
2. Block-Diagonal Darboux Form: There exists a Darboux basis $\{u_1,\dots,u_n,w_1,\dots,w_n\}$ in which the matrix of $f$ is block-diagonal:

$$[f] = \begin{pmatrix} B & 0 \ 0 & B^T \end{pmatrix}, \quad B\in M_n(K).$$

1. Jordan Form: If all eigenvalues of $f$ are in the base field, $B$ can be chosen in Jordan normal form, i.e., $$A = \operatorname{diag}( J_1,\dots,J_r; J_1^T,\dots,J_r^T )$$ with $J_i$ Jordan blocks.

This property underlies a fully elementary yet complete proof of the symplectic spectral theorem for self-adjoint operators, providing a symplectic analogue to the classical spectral theorem for (real or complex) inner product spaces.

3. Key Theorems and Structural Consequences

Several fundamental theorems and corollaries flow from the SSSP, particularly in the context of solving symplectic eigenvalue problems for sparse matrices associated to graphs (Gupta et al., 17 Jan 2026):

• Supergraph Theorem: If $N$ satisfies SSSP for $G^L$, then for any supergraph $H^L\supseteq G^L$, there exists $N'\in S(H^L)$ with the same symplectic eigenvalues and SSSP. This allows unimpeded graph extension without loss of spectral data.
• Bifurcation Theorem: If $N$ has SSSP, then in any sufficiently small neighborhood, every positive definite symmetric perturbation can be matched, via adjustment, to a matrix on the same graph with identical symplectic spectrum.
• Multiplicity Refinement: SSSP implies that every refinement of the symplectic spectral multiplicity partition is also realizable on the same graph.
• Matrix Liberation Lemma: Given $N$ with SSSP on $G^L$ and a direction $R$, if SSSP holds with respect to new edges in the sparsity induced by $R$, precise control over edge insertion is possible while preserving spectrum.

These results constitute a framework for constructing and adjusting sparse positive definite matrices with prescribed symplectic spectra.

4. Algorithmic Verification and Construction

SSSP can be checked and exploited algorithmically via matrix rank conditions and explicit construction methods (Gupta et al., 17 Jan 2026):

• Verification: Form the verification matrix $\Xi(N)$ associated with non-edges of $G^L$ and check for row rank sufficiency (Theorem 5.6). Alternatively, assemble the tangent-plus-sparsity matrix $\Phi(N)$ and verify the relevant submatrix rank.
• Construction: Start with diagonal $D\oplus D$ (for given simple symplectic eigenvalues), then use the Supergraph Theorem to add desired edges through controlled Hamiltonian perturbations and local solutions to the appropriate equations. Linearization (via the Inverse Function Theorem) and Newton-type iterative corrections are used in practical realization.
• Graph Couplings and Zero-Forcing: Structural and combinatorial techniques (such as coupled zero-forcing) contribute to understanding maximum multiplicities and reducing labeling complexity.

5. Worked Examples and Explicit Forms

A prototypical illustration involves the labeled bipartite graph $K_{2,2}^M$, with SSSP verified explicitly for

$$N = \begin{pmatrix} 1 & 0 & -\frac{1}{\sqrt2} & \frac{1}{\sqrt2} \ 0 & 1 & \frac{1}{\sqrt2} & \frac{1}{\sqrt2} \ -\frac{1}{\sqrt2} & \frac{1}{\sqrt2} & 2 & 0 \ \frac{1}{\sqrt2} & \frac{1}{\sqrt2} & 0 & 2 \end{pmatrix}$$

via the verification matrix $\Xi(N)$ having full rank. Its symplectic eigenvalues are both 1, and the Supergraph Theorem guarantees spectrum preservation for all supergraphs.

This explicit mode of demonstration aligns with the block-diagonal and polarization constructs for self-adjoint operators, where normal forms, including $[f]=\operatorname{diag}(J, J^T)$, are realized by synthesis of suitable Darboux bases and operator projections (Malagón, 2017).

6. Additional Structural Insights and Corollaries

Important structural insights derived from SSSP include:

• Sparsity Lower Bound: Any irreducible $N\succ0$ with equal symplectic eigenvalues has at least $4n-4$ nonzero entries (or $3p-2$ edges) [(Gupta et al., 17 Jan 2026), Corollary 3.5]. The triangular-path graph family attains this minimum.
• Direct Sum and Disconnected Union: Symplectic spectra compose over disconnected unions and direct sums. The direct-sum-plus-SSSP lemma states that blocks with non-overlapping symplectic spectra can be combined into larger SSSP blocks (Theorem 7.5).
• Complete Characterization for $n=4$: For order-$4$ graphs, SSSP fully governs the realization of all possible symplectic spectra, supplementing with a finite number of ad hoc checks.
• Galois Descent: In the context of algebraic fields, normal forms and SSSP-based decompositions can be constructed over field extensions and then descended to the base field using Galois theory [(Malagón, 2017), Proposition 9].

Consequence	Description	Reference
Supergraph Theorem	Spectrum preservation under edge addition	(Gupta et al., 17 Jan 2026)
Sparsity Lower Bound	Minimum nonzeros for irreducible matrices with equal eigenvalues	(Gupta et al., 17 Jan 2026)
Direct Sum Lemma	SSSP blocks compose when spectra do not overlap	(Gupta et al., 17 Jan 2026)

7. Connections and Broader Impact

The SSSP bridges the theory of Hamiltonian and symplectic linear algebra with constructive combinatorial matrix theory, underpinning both the spectral theory of operators on symplectic spaces and the realization problem for symplectic eigenvalues under graph sparsity. Its equivalent formulations in terms of matrix spanning, Hadamard-sparsity, and commutation with the symplectic form, provide a suite of analytical, constructive, and algorithmic tools. The property is fundamental for classification of canonical forms, synthesis of matrices with prescribed invariants, and for the design of flexible constructions in graphical symplectic eigenvalue problems, with ramifications in symplectic geometry, integrable systems, quantum information, and applied mathematics (Malagón, 2017, Gupta et al., 17 Jan 2026).