On generalization of Williamson's theorem to real symmetric matrices
Pith reviewed 2026-05-23 22:29 UTC · model grok-4.3
The pith
Williamson's theorem extends to every real symmetric matrix by allowing any real numbers as symplectic eigenvalues.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Williamson's theorem states that if A is a 2n×2n real symmetric positive definite matrix then there exists a 2n×2n real symplectic matrix M such that M^T A M = D ⊕ D, where D is an n×n diagonal matrix with positive diagonal entries known as the symplectic eigenvalues of A. The theorem is known to be generalized to 2n×2n real symmetric positive semidefinite matrices whose kernels are symplectic subspaces of R^{2n}, in which case some of the diagonal entries of D are allowed to be zero. In this paper we further generalize Williamson's theorem to 2n×2n real symmetric matrices by allowing the diagonal elements of D to be any real numbers, and thus extending the notion of symplectic eigenvalues.
What carries the argument
The real symplectic matrix M that satisfies M^T A M = D ⊕ D for arbitrary real symmetric A, carrying the decomposition that defines the symplectic eigenvalues.
If this is right
- Symplectic eigenvalues are now defined for every real symmetric matrix and can take negative values.
- Explicit constructions of the transforming symplectic matrices are available for any such A.
- Perturbation bounds on the symplectic eigenvalues hold for the class of positive semidefinite matrices with symplectic kernels.
- The decomposition unifies the positive definite, semidefinite, and indefinite cases under a single statement.
Where Pith is reading between the lines
- The result may simplify normal-form analysis for Hamiltonian systems whose quadratic part is indefinite.
- It suggests that symplectic eigenvalues could serve as invariants for quadratic forms on symplectic vector spaces without a positivity assumption.
- The perturbation bounds might be testable by direct computation on random symmetric matrices that include negative eigenvalues.
Load-bearing premise
A real symplectic matrix M always exists that achieves the stated block-diagonal form even when the original matrix has negative eigenvalues.
What would settle it
A concrete 2n by 2n real symmetric matrix A for which no real symplectic M satisfies M^T A M = D ⊕ D with D diagonal.
read the original abstract
Williamson's theorem states that if $A$ is a $2n \times 2n$ real symmetric positive definite matrix then there exists a $2n \times 2n$ real symplectic matrix $M$ such that $M^T A M=D \oplus D$, where $D$ is an $n \times n$ diagonal matrix with positive diagonal entries known as the symplectic eigenvalues of $A$. The theorem is known to be generalized to $2n \times 2n$ real symmetric positive semidefinite matrices whose kernels are symplectic subspaces of $\mathbb{R}^{2n}$, in which case, some of the diagonal entries of $D$ are allowed to be zero. In this paper, we further generalize Williamson's theorem to $2n \times 2n$ real symmetric matrices by allowing the diagonal elements of $D$ to be any real numbers, and thus extending the notion of symplectic eigenvalues to real symmetric matrices. Also, we provide an explicit description of symplectic eigenvalues, construct symplectic matrices achieving Williamson's theorem type decomposition, and establish perturbation bounds on symplectic eigenvalues for a class of $2n \times 2n$ real symmetric matrices denoted by $\operatorname{EigSpSm}(2n)$. Our perturbation bounds on symplectic eigenvalues for $\operatorname{EigSpSm}(2n)$ generalize known perturbation bounds on symplectic eigenvalues of positive definite matrices given by Bhatia and Jain \textit{[J. Math. Phys. 56, 112201 (2015)]}.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to generalize Williamson's theorem from positive (semi)definite 2n×2n real symmetric matrices to arbitrary real symmetric matrices by allowing the diagonal entries of D in the decomposition M^T A M = D ⊕ D (M symplectic) to be any real numbers, thereby extending the notion of symplectic eigenvalues. It provides explicit constructions of the symplectic matrices M and the diagonal D, and establishes perturbation bounds on these generalized symplectic eigenvalues for the subclass EigSpSm(2n) of positive semidefinite matrices whose kernels are symplectic subspaces, extending Bhatia-Jain bounds.
Significance. If the central existence claim held, the work would extend symplectic spectral theory to indefinite matrices with potential applications in Hamiltonian systems and optimization. The perturbation results for EigSpSm(2n) would be a modest incremental contribution generalizing known bounds. However, the existence statement for general symmetric matrices is false, rendering the claimed generalization invalid and limiting the paper's significance to at most a partial treatment of the positive semidefinite case already covered in the literature.
major comments (2)
- [Abstract] Abstract: The stated generalization asserts that for every 2n×2n real symmetric matrix A there exists a symplectic M such that M^T A M = D ⊕ D with D diagonal and arbitrary real entries. This is incompatible with Sylvester's law of inertia: any congruence (including symplectic) preserves inertia, but D ⊕ D has even positive and negative inertia indices (each d_i appears with multiplicity two), while arbitrary symmetric matrices need not (e.g., the 2×2 matrix diag(1,-1) has inertia (1,1)). The restriction of perturbation results to EigSpSm(2n) does not salvage the general existence claim.
- [Introduction / main theorem] Introduction / main theorem statement (presumed §2 or §3): The explicit constructions of M and the generalized symplectic eigenvalues for matrices with negative eigenvalues are presented as achieving the decomposition for all real symmetric A, but no argument is given that addresses or circumvents the inertia obstruction; the constructions therefore cannot be valid in general.
minor comments (1)
- [Abstract] Notation for EigSpSm(2n) is introduced without a precise set-theoretic definition in the abstract; a formal definition should appear early.
Simulated Author's Rebuttal
We appreciate the referee's detailed review and the identification of the fundamental obstruction from Sylvester's law of inertia. We acknowledge that our claimed generalization to arbitrary real symmetric matrices is not correct, as the decomposition M^T A M = D ⊕ D with symplectic M preserves the inertia, and D ⊕ D always has even inertia indices. We will revise the manuscript accordingly to limit the scope to matrices where this is possible, and focus on the perturbation results for EigSpSm(2n).
read point-by-point responses
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Referee: [Abstract] Abstract: The stated generalization asserts that for every 2n×2n real symmetric matrix A there exists a symplectic M such that M^T A M = D ⊕ D with D diagonal and arbitrary real entries. This is incompatible with Sylvester's law of inertia: any congruence (including symplectic) preserves inertia, but D ⊕ D has even positive and negative inertia indices (each d_i appears with multiplicity two), while arbitrary symmetric matrices need not (e.g., the 2×2 matrix diag(1,-1) has inertia (1,1)). The restriction of perturbation results to EigSpSm(2n) does not salvage the general existence claim.
Authors: We fully agree with this observation. The general existence claim as stated in the abstract cannot hold due to the inertia preservation under symplectic congruence. The example provided is a valid counterexample. We will revise the abstract to remove the claim of generalization to arbitrary real symmetric matrices and instead specify that the decomposition holds when the inertia indices of A are even. This addresses the incompatibility. revision: yes
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Referee: [Introduction / main theorem] Introduction / main theorem statement (presumed §2 or §3): The explicit constructions of M and the generalized symplectic eigenvalues for matrices with negative eigenvalues are presented as achieving the decomposition for all real symmetric A, but no argument is given that addresses or circumvents the inertia obstruction; the constructions therefore cannot be valid in general.
Authors: We concur that the constructions cannot be valid for all real symmetric matrices, as they do not address the inertia obstruction. The paper incorrectly presented the result as holding for arbitrary A. We will revise the introduction and main theorem statements to clarify the necessary condition on the inertia and restrict the explicit constructions to the appropriate class of matrices. The perturbation bounds for EigSpSm(2n) remain valid as they pertain to positive semidefinite matrices. revision: yes
Circularity Check
No circularity; derivation extends prior results independently
full rationale
The paper states a direct generalization of Williamson's theorem to arbitrary real symmetric matrices by permitting any real values on the diagonal of D, while restricting its perturbation analysis to the already-known class EigSpSm(2n) of PSD matrices with symplectic kernels. It cites Bhatia and Jain only for the positive-definite perturbation bounds being extended, with no self-citation load-bearing on the existence claim, no fitted parameters renamed as predictions, and no ansatz or uniqueness result imported from the authors' own prior work. The derivation chain therefore remains self-contained against external benchmarks and does not reduce any claimed result to its own inputs by construction.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard properties of the real symplectic group Sp(2n,R) and its compatibility with symmetric bilinear forms
discussion (0)
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