---
title: Symplectic Eigenvalues
url: https://www.emergentmind.com/topics/symplectic-eigenvalues
type: topic
---

# Symplectic Eigenvalues

A symplectic eigenvalue is a canonical invariant arising from the diagonalization of real positive-definite matrices (and certain classes of operators) under congruence by symplectic transformations, as formalized by Williamson’s theorem. Symplectic eigenvalues encode structure that is central in areas ranging from continuous-variable quantum information theory to classical Hamiltonian systems, operator theory, majorization theory, and Riemannian geometry on matrix spaces. This entry synthesizes recent developments encompassing finite and infinite-dimensions, variational and majorization principles, optimization, sensitivity theory, and connections to operator inequalities.

## 1. Definition, Characterizations, and Williamson’s Theorem

Let $A \in \mathbb{R}^{2n \times 2n}$ be symmetric positive-definite and let $J := \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}$. The real symplectic group $\mathrm{Sp}(2n)$ consists of those $S$ with $S^T J S = J$.

**Williamson’s Theorem (1936):** There exists $S \in \mathrm{Sp}(2n)$ and $D = \mathrm{diag}(d_1, \ldots, d_n)$ with $d_1 \leq \cdots \leq d_n > 0$ such that
$$
S^T A S = \begin{pmatrix} D & 0 \\ 0 & D \end{pmatrix}.
$$
The unique positive numbers $d_1, \ldots, d_n$ are the **symplectic eigenvalues** of $A$ [1803.04647][2506.07560].

Equivalent characterizations include:
- $iJA$ has eigenvalues $\pm d_1, \ldots, \pm d_n$.
- $i A^{1/2} J A^{1/2}$ is Hermitian with spectrum $\{\pm d_1, \ldots, \pm d_n\}$ [2007.10572].
- Each symplectic eigenvalue $d_i$ appears with multiplicity two in the spectrum of $A$.
- For positive-semidefinite $A$, the notion generalizes if $\ker A$ is a symplectic subspace [2208.05291].

Symplectic eigenvectors come in pairs: $A u_i = d_i J v_i$, $A v_i = -d_i J u_i$ with $u_i^T J v_j = \delta_{ij}$.

In the **operator-theoretic infinite-dimensional setting**, if $T$ is a positive, invertible operator on $\mathcal{H} \oplus \mathcal{H}$ (with $T - \alpha I$ compact), the symplectic spectrum of $T$ consists of the positive eigenvalues of $M := i \sqrt{T} J \sqrt{T}$ and is at most countable, accumulating only at $\alpha$ [2212.03900][2303.09309].

## 2. Variational Principles and Optimization

Symplectic eigenvalues support variational characterizations mirroring classical eigenvalue theory:
- **Wielandt-Type Min-Max Principle:** The sum of selected symplectic eigenvalues is given by a max-min formula over symplectically orthonormal pairs in nested subspaces [2108.10741].
- **Trace-Minimization Theorem:** For $k \leq n$,
  $$
  \sum_{j=1}^k d_j(A) = \min_{S \in \mathrm{Sp}(2n, 2k)} \frac{1}{2} \operatorname{tr}(S^T A S),
  $$
  where $\mathrm{Sp}(2n, 2k)$ is the symplectic Stiefel manifold, and the minimum is achieved by symplectic eigenvector sets [2101.02618][2506.07560][2208.05291].
- Critical points of the Brockett-type or trace objectives under symplecticity constraints correspond to symplectic eigenvectors, and Riemannian optimization methods (e.g., trust-region, gradient descent on symplectic Stiefel manifolds) are effective for large-scale computation [2101.02618][2506.07560].

For infinite-dimensional operators, finite-rank truncations converge to the symplectic spectrum under mild hypotheses, with explicit control available for certain operator classes (e.g., block diagonal or block circulant forms) [2303.09309].

## 3. Inequalities: Majorization, Interlacing, and Schur–Horn Theory

Fundamental inequalities for symplectic eigenvalues have close analogs to—but often crucial differences from—those for ordinary Hermitian eigenvalues.

### Majorization and Supermajorization

For $A \in \mathbb{R}^{2n \times 2n}$ positive-definite, let $d(A) = (d_1(A),...,d_n(A))$:
- **Weak Supermajorization (Schur–Horn for Symplectic Eigenvalues):**
  $$
  \Delta^s(A) \prec^w d(A),
  $$
  where $\Delta^s_j(A) = \sqrt{[A_{11}]_{jj} [A_{22}]_{jj}}$ and blocks are as in $A = \begin{pmatrix} A_{11} & A_{12} \\ A_{12}^T & A_{22} \end{pmatrix}$ [2004.03906][2404.05795].
- For fixed $y \in \mathbb{R}^n_{>0}$, all symplectic diagonals $x$ arise from $d(A) = y$ if and only if $x \prec^w y$.
- Full majorization $\prec$ (i.e., equality in all partial sums) holds precisely when $A$ is orthosymplectically diagonalizable [2404.05795].

### Interlacing with Ordinary Eigenvalues

For $A$ real positive-definite with eigenvalues $\lambda_1 \leq \cdots \leq \lambda_{2n}$,
$$
\lambda_j(A) \leq d_j(A) \leq \lambda_{n+j}(A),
$$
and $d(A) \prec_{\log} (\lambda_1(A),..., \lambda_{2n}(A))$ [1803.04647].

### Sums and Products—Multiplicative and Additive Lidskii Inequalities

For $A,B > 0$,
- **Symplectic Lidskii Inequality:** For any $k$ and index sets,
  $$
  \sum_{r=1}^k d_{i_r}(A+B) \geq \sum_{r=1}^k d_{i_r}(A) + \sum_{r=1}^k d_r(B),
  $$
  with equality if and only if certain common symplectic eigenvector conditions hold [2309.04562][2004.11024].
- **Multiplicative Inequalities:** For geometric means,
  $$
  d(A \#_t B) \prec_{\log} d(A)^{1-t} d(B)^t,
  $$
  and for Riemannian/Cartan barycenters (geometric means of random matrices),
  $$
  \log d(G(\mu)) \prec_w \int \log d(A) \, d\mu(A).
  $$
  [1710.00494][1803.04647][2108.10741]

### Horn-Type and Trace Inequalities

The Horn cone for symplectic eigenvalues satisfies all classical Horn inequalities, except trace equality $\operatorname{Tr}(C) = \operatorname{Tr}(A) + \operatorname{Tr}(B)$ is replaced by $\operatorname{Tr}(C) \geq \operatorname{Tr}(A) + \operatorname{Tr}(B)$ [2202.10260].

## 4. Analyticity, Sensitivity, and Differential Properties

- **Continuity:** The maps $A \mapsto d_j(A)$ are continuous on the positive-definite cone.
- **Differentiability:** The functions $d_m(A)$ are not everywhere differentiable (unless the eigenvalue is simple), but always directionally differentiable. If $d_m(A)$ is simple:
  $$
  D d_m(A)[E] = -\frac{1}{2} \langle x x^T + y y^T, E \rangle,
  $$
  where $(x, y)$ is the corresponding normalized symplectic eigenpair of $A$ [2007.10572][2004.11024].
- **Subdifferential Structure:** Clarke and Michel–Penot subdifferentials of $-d_m$ are convex hulls of matrices of the form $-\frac{1}{2}(x x^T + y y^T)$ for all associated symplectic eigenpairs [2007.10572].
- **Analytic Paths:** Along $t \mapsto A(t)$ real-analytic, symplectic eigenvalues are piecewise real-analytic; their derivatives obey explicit Rayleigh-quotient-type formulas [2004.11024].

## 5. Infinite-Dimensional Extensions

A class of infinite-dimensional positive, invertible operators $T: \mathcal{H} \oplus \mathcal{H} \rightarrow \mathcal{H} \oplus \mathcal{H}$, with $T-\alpha I$ compact, admits a symplectic spectrum via the spectrum of $i\sqrt{T} J \sqrt{T}$. Main results include:
- Symplectic eigenvalues to the right (resp. left) of $\alpha$ are always less than (resp. greater than) the corresponding ordinary eigenvalues [2212.03900].
- For infinite-mode bosonic Gaussian covariance operators, symplectic eigenvalues determine entropic and operational quantities; finite-rank truncations converge to the true spectrum under mild assumptions [2303.09309].
- The symplectic Szegő theorem gives limiting distributions for symplectic spectra of block Toeplitz operators and provides entropy rates for stationary quantum Gaussian processes [2006.11829].

## 6. Applications, Numerical Aspects, and Generalizations

Symplectic eigenvalues are indispensable in quantum optics, continuous-variable quantum information (notably for Gaussian states), Hamiltonian dynamics, operator theory, and sensitivity analysis. Key contexts include:
- **Gaussian Covariance Matrices:** A $2n \times 2n$ covariance matrix is physical if and only if all symplectic eigenvalues satisfy $d_j \geq \frac{1}{2}$, with explicit entropic formulas and operational relevance [1803.04647][2303.09309][2006.11829].
- **Fisher Information:** Symplectic decomposition provides insight into sensitivity for parameter pairs (e.g., $(q_j, p_j)$), complementing ordinary eigenanalysis [2207.12077].
- **Optimization:** Riemannian and symplectic-structure-preserving algorithms target minimal symplectic eigenvalues and eigenvectors, with competitive computational properties [2101.02618][2506.07560].
- **Log-Majorization and Cartan Means:** Symplectic eigenvalue maps are monotone, homogeneous, and Lipschitz for trace metric; log-majorization bounds are preserved under Riemannian/Cartan barycenters [1710.00494].

## 7. Open Problems and Outlook

Notable open directions include:
- Generalization of interlacing and majorization inequalities to broader or singular infinite-dimensional settings (e.g., non-invertible operators, uncountable spectrum).
- Characterization of equality cases for symplectic-analytic inequalities beyond Schur–Horn, Ky Fan, and Lidskii types [2404.05795][2309.04562].
- Deeper understanding of variational and entropy-like principles for symplectic spectra in non-classical or infinite-component systems.
- Extension of truncation and approximation methods to new classes of infinite-dimensional quantum systems [2212.03900][2303.09309].

Symplectic eigenvalues thus constitute a rich, unifying invariant at the interface of algebraic, analytic, and geometric analysis, with ongoing developments in optimization, infinite-dimensional analysis, and quantum information theory.

Source: https://www.emergentmind.com/topics/symplectic-eigenvalues