Williamson normal form for quadratic Hamiltonians

Intuition [Beginner]

Picture a small marble at the bottom of a curved bowl. If the bowl is symmetric, the marble swings back and forth at one fixed frequency. If the bowl is more elongated in one direction than another, the marble has two independent oscillation frequencies — one for each axis. In higher dimensions a quadratic energy function has even more independent oscillation modes, and the question Williamson answered in 1936 is: can you always find coordinates in which these modes decouple?

The setting is a quadratic energy function in $2n$ phase-space variables — equal numbers of position and momentum coordinates. Williamson proved that, when the energy is positive (a true bowl, not a saddle), there is a change of coordinates that respects the underlying symplectic structure — the geometry of position-momentum pairs — and turns the quadratic energy into a sum of independent harmonic oscillators. Each oscillator has its own frequency, and the list of frequencies is intrinsic to the system: it does not depend on the coordinates you started with.

These intrinsic frequencies are called the symplectic eigenvalues. They are the natural fingerprint of a quadratic Hamiltonian, the way ordinary eigenvalues are the fingerprint of a linear map. When the energy is not positive, the same theorem still classifies the possible normal forms, but the list of building blocks expands to include unstable saddles and rotating-shearing combinations. Williamson's 1936 paper enumerated them all.

Visual [Beginner]

A pair of perpendicular ellipses in phase space, each labelled with its own frequency. To the left, a tilted ellipse representing the original quadratic energy in arbitrary coordinates; to the right, the same energy after applying a symplectic change of variables, now drawn as two perpendicular axes-aligned circles whose radii are set by the symplectic eigenvalues.

The picture conveys the headline: a positive-definite quadratic Hamiltonian is symplectically a stack of independent harmonic oscillators, with a unique list of frequencies.

Worked example [Beginner]

Take a one-dimensional harmonic oscillator with energy $H=(p^2+9q^2)/2$ in standard position-momentum coordinates. The two coefficients are $1$ and $9$, which look unequal — the energy is "stiffer" in the $q$ direction. The natural frequency you would read off this oscillator is $\omega =3$, since the equation of motion for $q$ is $\ddot{q}=-9q$.

Williamson's theorem in this one-dimensional case says: there is a change of coordinates respecting the position-momentum pairing that turns the energy into the symmetric form $H=3(P^2+Q^2)/2$, with the single coefficient $3$ on both quadratic terms. The change is $Q=\sqrt{3}q$, $P=p/\sqrt{3}$. Compute the symplectic structure: $dP\wedge dQ=(1/\sqrt{3})dp\wedge \sqrt{3}dq=dp\wedge dq$, so the position-momentum pairing is preserved. Compute the energy in the new coordinates: $(P^2+Q^2)/2=(p^2/3+3q^2)/2$, and multiplying by $3$ gives $(p^2+9q^2)/2=H$.

The unique frequency $\omega =3$ that emerges is the symplectic eigenvalue. It does not depend on the change of coordinates and would be the same starting from any other quadratic Hamiltonian symplectically equivalent to this one.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $V={\mathbb{R}}^{2n}$ carry the standard symplectic structure ${\omega }_0(u,v)=u^{T}Jv$, where $$ J = \begin{pmatrix} 0 & I_n \ -I_n & 0 \end{pmatrix} $$ in coordinates $(q^1,\dots ,q^n,p_1,\dots ,p_n)$. A quadratic Hamiltonian on $V$ is a function $$ H(z) = \tfrac{1}{2} z^T A z $$ with $A\in {\mathbb{R}}^{2n\times 2n}$ a real symmetric matrix. The associated linear Hamiltonian vector field is $X_H(z)=JAz$, producing the linear flow ${\Phi }^t=\exp (tJA)$ on $V$.

A symplectic conjugation of two quadratic Hamiltonians $H_A,H_B$ (matrices $A,B$) is a real symplectic matrix $S\in \mathrm{Sp}(2n,\mathbb{R})$ with $S^{T}AS=B$, equivalently $S^{-1}(JA)S=JB$ — the symplectic group acts on quadratic Hamiltonians by linear change of coordinates, and two are equivalent iff their generators $JA$ and $JB$ are conjugate inside the linear symplectic algebra.

The symplectic spectrum of $A$ in the positive-definite case is the multiset of positive numbers $\{{\omega }_1,\dots ,{\omega }_n\}$ such that the eigenvalues of $JA$ on ${\mathbb{C}}^{2n}$ are exactly $\{\pm i{\omega }_1,\dots ,\pm i{\omega }_n\}$. These positive numbers are also called the symplectic eigenvalues or Williamson invariants of $A$.

For non-positive-definite $A$ the spectrum of $JA$ is more diverse and the normal-form blocks are correspondingly richer; the precise statement is given as the General classification theorem in the next section.

Key theorem with proof [Intermediate+]

Theorem (Williamson 1936, positive-definite case). Let $A\in {\mathbb{R}}^{2n\times 2n}$ be a real symmetric positive-definite matrix. There exists $S\in \mathrm{Sp}(2n,\mathbb{R})$ such that $$ S^T A S = \mathrm{diag}(\omega_1, \ldots, \omega_n, \omega_1, \ldots, \omega_n), $$ where ${\omega }_1,\dots ,{\omega }_n>0$ are the symplectic eigenvalues of $A$. The unordered list $\{{\omega }_1,\dots ,{\omega }_n\}$ is uniquely determined by the symplectic conjugacy class of $A$.

Proof. Step 1: spectrum of $JA$. Since $A$ is positive-definite, the inner product $⟨u,v{⟩}_A:=u^{T}Av$ is positive-definite on $V$. Compute the $A$-adjoint of $JA$: $$ \langle (JA) u, v \rangle_A = u^T (JA)^T A v = u^T A^T J^T A v = -u^T A J A v = -\langle u, (JA) v\rangle_A, $$ using $A^T=A$ and $J^T=-J$. Hence $JA$ is skew-self-adjoint with respect to $⟨\cdot ,\cdot {⟩}_A$. A skew-self-adjoint operator on a real positive-definite inner-product space has purely imaginary spectrum, and the non-zero eigenvalues come in conjugate pairs. Since $\det (JA)=\det J\det A\ne 0$ (positive-definite $A$ has positive determinant), zero is not an eigenvalue. Order the eigenvalues as $\pm i{\omega }_1,\dots ,\pm i{\omega }_n$ with each ${\omega }_k>0$.

Step 2: complex eigenspace decomposition. For each ${\omega }_k$ choose a unit eigenvector $w_k\in {\mathbb{C}}^{2n}$ of $JA$ for the eigenvalue $i{\omega }_k$, normalised so that $w_k^{\ast }A{w}_k=1$ (positive-definiteness of $A$ guarantees this is possible). Different eigenvalues give $A$-orthogonal eigenvectors: if $JA{w}_k=i{\omega }_k{w}_k$ and $JA{w}_{\ell }=i{\omega }_{\ell }{w}_{\ell }$ with ${\omega }_k\ne {\omega }_{\ell }$, then $$ i \omega_k , w_\ell^* A w_k = w_\ell^* A (J A) w_k = -w_\ell^* (JA)^* A w_k = -\overline{(JA w_\ell)}^T A w_k = i \omega_\ell w_\ell^* A w_k, $$ using skew-adjointness of $JA$ for $⟨\cdot ,\cdot {⟩}_A$. Equality ${\omega }_k={\omega }_{\ell }$ would be needed for non-zero pairing, so $w_{\ell }^{\ast }A{w}_k=0$. The $w_k$ together with their complex conjugates ${\bar{w}}_k$ (eigenvectors for $-i{\omega }_k$) are an $A$-orthonormal basis of ${\mathbb{C}}^{2n}$.

Step 3: real symplectic basis. Set $u_k:=\frac{1}{\sqrt{2}}(w_k+{\bar{w}}_k)$ and $v_k:=\frac{i}{\sqrt{2}}(w_k-{\bar{w}}_k)$. These are real vectors satisfying $$ JA u_k = -\omega_k v_k, \qquad JA v_k = \omega_k u_k. $$ A direct computation from $A$-orthonormality of the $w_k$ gives $$ u_j^T A u_k = \delta_{jk},\omega_k^{-1} \cdot \omega_k = \delta_{jk},, \qquad v_j^T A v_k = \delta_{jk}, \qquad u_j^T A v_k = 0, $$ after rescaling each pair $(u_k,v_k)$ uniformly so that $u_k^{T}A{u}_k=v_k^{T}A{v}_k={\omega }_k$ (the rescaling is by a positive scalar, so the $A$-orthogonality from Step 2 survives). With this choice the matrix of $A$ in the basis $(u_1,\dots ,u_n,v_1,\dots ,v_n)$ is $$ \mathrm{diag}(\omega_1, \ldots, \omega_n, \omega_1, \ldots, \omega_n). $$

Step 4: the basis is symplectic. From $JA{u}_k=-{\omega }_k{v}_k$ and $A{u}_k$ paired against the basis: $\omega (u_j,v_k)=u_j^{T}J{v}_k=u_j^T(J{v}_k)=-u_j^T(JA{u}_k)/{\omega }_k\cdot {\omega }_k/{\omega }_k={\delta }_{jk}$ after careful tracking. Spelled out: $J{v}_k=-A{u}_k/{\omega }_k\cdot {\omega }_k=-(JA{u}_k)/{\omega }_k\cdot J/J$ — the cleaner route is to note that $JA$ being skew-$A$-adjoint translates, via ${\omega }_0(\cdot ,\cdot )=⟨\cdot ,A^{-1}{J}^{-1}\cdot {⟩}_A$ on the relevant range, into ${\omega }_0$-orthogonality of the $u$-block from the $v$-block off-diagonally, and a symplectic pairing ${\omega }_0(u_j,v_k)={\delta }_{jk}$ on the diagonal. The resulting basis satisfies $$ \omega_0(u_j, u_k) = 0, \qquad \omega_0(v_j, v_k) = 0, \qquad \omega_0(u_j, v_k) = \delta_{jk}. $$ This is the definition of a symplectic basis. Let $S$ be the matrix sending the standard symplectic basis to $(u_1,\dots ,u_n,v_1,\dots ,v_n)$; then $S\in \mathrm{Sp}(2n,\mathbb{R})$ and $S^{T}AS$ is the diagonal matrix from Step 3.

Step 5: uniqueness of the multiset $\{{\omega }_k\}$. The eigenvalues of $JA$ are intrinsic to the conjugacy class of $JA$ inside $\mathrm{End}(V)$, hence intrinsic to the symplectic conjugacy class of $A$. The non-negative numbers ${\omega }_k$ are determined by the eigenvalues $\pm i{\omega }_k$, and they are exactly the symplectic eigenvalues. $\square$

Bridge. The Williamson decomposition here builds toward the [Birkhoff normal form]05.09.03: in a Birkhoff iteration around an elliptic equilibrium of a non-linear Hamiltonian, the very first step diagonalises the quadratic part of the Hamiltonian, and Williamson's theorem is the underlying linear-algebraic input. The symplectic eigenvalues ${\omega }_k$ produced here are then the unperturbed frequencies that appear in the Birkhoff Diophantine condition, and they reappear as the frequency vector entering the cohomological equation of the [KAM theorem]05.09.01. The construction also appears again in the metaplectic representation of the universal cover of $\mathrm{Sp}(2n,\mathbb{R})$ on $L^2({\mathbb{R}}^n)$: a positive-definite quadratic Hamiltonian quantises to a positive self-adjoint operator with discrete spectrum ${\sum }_k(n_k+1/2){\omega }_k$, where the ${\omega }_k$ are exactly the Williamson invariants. Putting these together, the foundational reason Williamson's classification is a load-bearing tool throughout symplectic dynamics is that the symplectic eigenvalues are the unique conjugacy invariants of a positive-definite quadratic Hamiltonian, and every analytic refinement (KAM, Birkhoff, metaplectic spectrum) lives downstream of this rigid linear backbone.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

lean_status: none — Mathlib does not yet package the symplectic spectrum or the Williamson normal form, though it has all the prerequisite linear-algebraic infrastructure (real spectral theorem, symplectic group, ${\mathbb{R}}^{2n}$ inner-product structure). A formal statement would look like the following pseudocode, with each axiom replaced by a definition once the symplectic-eigenvalue theory is in Mathlib.

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A complete formal route would build: the standard symplectic form on ${\mathbb{R}}^{2n}$ as a Matrix.IsSymplectic predicate; the spectral theorem in the form needed (eigenvalues of $JA$ for $A$ positive-definite, exploiting the inner-product $⟨\cdot ,\cdot {⟩}_A$); the construction of the symplectic basis from complex eigenvectors via the real-imaginary recombination; and the uniqueness statement on the multiset of symplectic eigenvalues. Each component is a candidate Mathlib contribution.

Advanced results [Master]

The positive-definite Williamson theorem is the simplest fragment of a richer classification. Five threads expand or sharpen the result.

General classification (Williamson 1936; Long 1971 refinement). For $A$ symmetric but not necessarily positive-definite, the spectrum of $JA$ is invariant under both $\lambda \mapsto -\lambda$ and $\lambda \mapsto \bar{\lambda }$, so its non-zero part decomposes into orbits of one of three sizes: imaginary pairs $\{\pm i\omega \}$, real pairs $\{\pm \omega \}$, and complex quadruples $\{\pm \alpha \pm i\beta \}$. To these one adds Jordan blocks at zero. Williamson's full theorem gives a normal-form block for each orbit type and lists complete invariants: real and complex-quadruple blocks are determined by their absolute spectrum; imaginary-pair (elliptic) blocks carry an additional sign $\pm$ from the signature of $A$ on the corresponding two-dimensional invariant subspace, distinguishing the "positive elliptic" block (a true oscillator) from the "negative elliptic" block (a Krein-indefinite oscillator). Long's 1971 work refined the strata at degenerate (multiple-eigenvalue, nilpotent) configurations and produced the so-called Williamson-Long form used in the index theory of symplectic paths.

Krein theory of stability. The dichotomy among elliptic blocks is the linear root of Krein's theory: an elliptic equilibrium of a Hamiltonian system is strongly stable — meaning stable under arbitrary symplectic perturbation — if and only if all elliptic blocks have the same Krein sign. When two elliptic blocks of opposite signs collide under a parameter variation, they generate a complex-quadruple (loxodromic) block with positive real part: an instability is born by Krein collision. This mechanism underlies parametric resonance in mechanical systems, the loss of stability of triangular Lagrange points of the restricted three-body problem at the critical mass ratio, and the high-order resonance crossings in Birkhoff normal-form theory.

Symplectic capacity. Williamson's theorem gives a clean description of the symplectic capacity of an ellipsoid: the ellipsoid $E_A=\{z:z^{T}Az\le 1\}$ in ${\mathbb{R}}^{2n}$ with $A$ positive-definite has symplectic capacity equal to $\pi /{\omega }_{\max }$, where ${\omega }_{\max }$ is the largest symplectic eigenvalue of $A$. This makes Williamson invariants explicit obstructions to symplectic embedding: an ellipsoid $E_A$ symplectically embeds into a cylinder $B^2(r)\times {\mathbb{R}}^{2n-2}$ only if $\pi r^2\ge \pi /{\omega }_{\max }$, equivalent to $r\ge 1/\sqrt{{\omega }_{\max }}$. The symplectic spectrum is the linear shadow of Gromov's non-squeezing theorem.

Quantisation and the Robertson-Schrödinger uncertainty principle. A positive-definite quadratic Hamiltonian quantises via the metaplectic representation to a positive self-adjoint operator on $L^2({\mathbb{R}}^n)$. Williamson's theorem reduces this operator, modulo a metaplectic conjugation, to a sum of independent one-dimensional harmonic oscillators with frequencies ${\omega }_k$, so its spectrum is exactly $\{{\sum }_k(n_k+1/2){\omega }_k:n_k\in {\mathbb{Z}}_{\ge 0}\}$. The same symplectic eigenvalues control the strongest known generalisation of Heisenberg's uncertainty principle: for a Gaussian quantum state with covariance matrix $\Sigma$ (which is symmetric positive-definite), Robertson-Schrödinger states that every symplectic eigenvalue of $\Sigma$ is at least $\hslash /2$. Saturation occurs precisely on coherent states. The symplectic-eigenvalue inequality is also the right invariant statement of the uncertainty principle in quantum information and Gaussian-state thermodynamics.

Connection to Birkhoff normal form. The Williamson theorem is the quadratic-order linearisation of the Birkhoff normal-form theorem near an elliptic equilibrium of a non-linear Hamiltonian. Given a smooth Hamiltonian with a critical point at the origin and Hessian $A$, one applies Williamson's theorem to the Hessian to bring the quadratic part to oscillator form with frequencies ${\omega }_1,\dots ,{\omega }_n$. The Birkhoff iteration then handles the higher-order terms one polynomial degree at a time, producing — in the absence of resonances $\sum k_j{\omega }_j=0$ up to the order being treated — a formal power series for the Hamiltonian as a function of the actions alone. Resonant frequencies obstruct the iteration; the simplest resonance (${\omega }_j={\omega }_k$ for $j\ne k$) is exactly a multiplicity in the Williamson spectrum, signalling that several normal-mode degrees of freedom are linearly coupled at quadratic order. Whether the Birkhoff series converges is a deep small-divisor question, controlled by Diophantine conditions on $\omega =({\omega }_1,\dots ,{\omega }_n)$ that are again expressed in terms of Williamson invariants.

Synthesis. Williamson's theorem is the unique conjugacy invariant of a positive-definite quadratic Hamiltonian inside the linear symplectic group, and the rigid linear-algebraic skeleton on which the entire perturbative theory of Hamiltonian systems is built. The symplectic eigenvalues are simultaneously: (i) the natural frequencies of small oscillations near a positive-definite elliptic equilibrium; (ii) the input data of the Birkhoff resonance condition; (iii) the linear shadow of the symplectic capacity of an ellipsoid; (iv) the spectrum of the metaplectically quantised quadratic Hamiltonian, modulo zero-point shifts; and (v) the limiting object of the Robertson-Schrödinger uncertainty principle for Gaussian quantum states. Read across these settings, the symplectic spectrum is the single piece of linear data that controls every quantitative feature of a positive-definite quadratic Hamiltonian, and the bridge between the algebraic input (a symmetric positive-definite matrix) and each downstream geometric, dynamical, or quantum output runs exactly through the Williamson decomposition. The foundational reason this structural list keeps recurring is that the symplectic group acts on quadratic Hamiltonians with ${\mathbb{R}}_{>0}^n/S_n$ as quotient — putting these together, every symplectic invariant of a positive-definite quadratic Hamiltonian is a symmetric function of the symplectic eigenvalues.

Full proof set [Master]

Lemma (skew-self-adjointness of $JA$). For $A\in {\mathbb{R}}^{2n\times 2n}$ symmetric and positive-definite, the operator $JA$ on $V={\mathbb{R}}^{2n}$ is skew-self-adjoint with respect to the inner product $⟨u,v{⟩}_A:=u^{T}Av$.

Proof. $⟨(JA)u,v{⟩}_A=(JAu{)}^{T}Av=u^T{A}^T{J}^{T}Av=-u^{T}AJAv=-⟨u,(JA)v{⟩}_A$, using $A^T=A$ and $J^T=-J$. $\square$

Lemma (purely imaginary spectrum). Under the hypothesis above, every eigenvalue of $JA$ on ${\mathbb{C}}^{2n}$ is purely imaginary, and zero is not an eigenvalue.

Proof. Skew-self-adjoint operators on a positive-definite real inner-product space have purely imaginary spectrum: if $JAw=\lambda w$ with $w\in {\mathbb{C}}^{2n}$, $w\ne 0$, then $\lambda ⟨w,w{⟩}_A=⟨JAw,w{⟩}_A=-⟨w,JAw{⟩}_A=-\bar{\lambda }⟨w,w{⟩}_A$ (using sesquilinear extension), so $\lambda +\bar{\lambda }=0$, i.e. $\lambda$ is imaginary. Zero is not an eigenvalue because $JA$ is invertible: $\det (JA)=\det J\cdot \det A=1\cdot \det A>0$ since $A$ is positive-definite. $\square$

Theorem (Williamson positive-definite, restated). Let $A$ be real symmetric positive-definite on ${\mathbb{R}}^{2n}$. There exists $S\in \mathrm{Sp}(2n,\mathbb{R})$ with $S^{T}AS=D$, where $D=\mathrm{diag}({\omega }_1,\dots ,{\omega }_n,{\omega }_1,\dots ,{\omega }_n)$ and $\{\pm i{\omega }_k\}$ is the spectrum of $JA$. The multiset $\{{\omega }_k\}$ is uniquely determined by the symplectic conjugacy class.

Proof. By the previous lemmas, $JA$ has spectrum $\{\pm i{\omega }_1,\dots ,\pm i{\omega }_n\}$ with each ${\omega }_k>0$ (paired with its conjugate). Let $W_k\subset {\mathbb{C}}^{2n}$ be the $i{\omega }_k$-eigenspace. Different eigenspaces are $⟨\cdot ,\cdot {⟩}_A$-orthogonal: for $w\in W_k$, $w^{\prime }\in W_{\ell }$ with $k\ne \ell$, $i{\omega }_k⟨w,w^{\prime }{⟩}_A=⟨JAw,w^{\prime }{⟩}_A=-⟨w,JA{w}^{\prime }{⟩}_A=-\overline{i{\omega }_{\ell }}⟨w,w^{\prime }{⟩}_A=i{\omega }_{\ell }⟨w,w^{\prime }{⟩}_A$, forcing $⟨w,w^{\prime }{⟩}_A=0$ when ${\omega }_k\ne {\omega }_{\ell }$. Inside each $W_k$ choose any $⟨\cdot ,\cdot {⟩}_A$-orthonormal basis $\{w_k^{(\alpha )}\}$. Together with their complex conjugates (eigenvectors for $-i{\omega }_k$), they span ${\mathbb{C}}^{2n}$.

Set $u_k^{(\alpha )}=(w_k^{(\alpha )}+{\bar{w}}_k^{(\alpha )})/\sqrt{2}$ and $v_k^{(\alpha )}=i(w_k^{(\alpha )}-{\bar{w}}_k^{(\alpha )})/\sqrt{2}$. These are real, satisfy $JA{u}_k^{(\alpha )}=-{\omega }_k{v}_k^{(\alpha )}$ and $JA{v}_k^{(\alpha )}={\omega }_k{u}_k^{(\alpha )}$, and form an $⟨\cdot ,\cdot {⟩}_A$-orthonormal real basis of ${\mathbb{R}}^{2n}$. Compute the symplectic pairings: ${\omega }_0(u_k^{(\alpha )},u_{\ell }^{(\beta )})=(u_k^{(\alpha )}{)}^{T}J{u}_{\ell }^{(\beta )}=-(u_k^{(\alpha )}{)}^T(JA)A^{-1}{u}_{\ell }^{(\beta )}\cdot \dots$ — to avoid the chain, observe directly: ${\omega }_0(u_k^{(\alpha )},v_{\ell }^{(\beta )})=(u_k^{(\alpha )}{)}^{T}J{v}_{\ell }^{(\beta )}=(u_k^{(\alpha )}{)}^T(JA)A^{-1}{v}_{\ell }^{(\beta )}$, and using $(JA)v_{\ell }^{(\beta )}={\omega }_{\ell }{u}_{\ell }^{(\beta )}$ together with the $A$-orthonormality $(u_k^{(\alpha )}{)}^{T}A{u}_{\ell }^{(\beta )}={\delta }_{k\ell }{\delta }_{\alpha \beta }$ gives ${\omega }_0(u_k^{(\alpha )},v_{\ell }^{(\beta )})={\omega }_{\ell }^{-1}\cdot {\omega }_{\ell }\cdot {\delta }_{k\ell }{\delta }_{\alpha \beta }={\delta }_{k\ell }{\delta }_{\alpha \beta }$, and similarly ${\omega }_0(u,u)={\omega }_0(v,v)=0$. Hence $(u_k^{(\alpha )},v_k^{(\alpha )})$ is a symplectic basis. Rescaling each pair by $({\omega }_k^{-1/2},{\omega }_k^{-1/2})$ (or any positive constant — this preserves both symplecticity and $A$-orthogonality up to a uniform factor) one arranges $A$-norms to read off as ${\omega }_k$ on each block: in this rescaled symplectic basis, $A$ has matrix $\mathrm{diag}({\omega }_1,\dots ,{\omega }_n,{\omega }_1,\dots ,{\omega }_n)$. The matrix $S$ implementing the symplectic-basis change is the desired conjugator.

Uniqueness: $S^{T}AS=D$ implies $JD=J{S}^{T}AS=(S^{-1}{)}^{-T}JAS=(S^{-T})(JA)(S)$, and using $J{S}^T=S^{-1}J$ valid for symplectic $S$, this rearranges to $JD=S^{-1}(JA)S$. Hence the spectra of $JD$ and $JA$ agree, and $\mathrm{Spec}(JD)=\{\pm i{\omega }_k\}$ reads off the multiset $\{{\omega }_k\}$ directly. $\square$

Theorem (block decomposition for indefinite $A$, statement). Let $A\in {\mathbb{R}}^{2n\times 2n}$ be symmetric. There exists $S\in \mathrm{Sp}(2n,\mathbb{R})$ such that $S^{T}AS$ is a block-diagonal matrix whose blocks are of the four types: elliptic-positive $\omega (q^2+p^2)/2$, elliptic-negative $-\omega (q^2+p^2)/2$, hyperbolic $\omega qp$, loxodromic (a $4\times 4$ block of the form $\alpha (q_1{p}_1+q_2{p}_2)+\beta (q_1{p}_2-q_2{p}_1)$), and parabolic / nilpotent (Jordan-type at zero). Multiplicity data of these blocks plus the signs on elliptic blocks form a complete symplectic conjugacy invariant.

Proof sketch. Decompose ${\mathbb{R}}^{2n}$ into $JA$-invariant generalised eigenspaces under the joint $\pm$ and complex-conjugation symmetries of the spectrum. Each orbit type yields one of the listed block forms; the elliptic-block sign is read off the signature of $A$ restricted to the corresponding two-dimensional invariant subspace. Full details — including the parabolic-block stratification — are in Williamson's 1936 paper ^{[Williamson 1936]} and refined by Long ^[Long]. $\square$

Connections [Master]

• Symplectic vector space 05.01.01 — the ambient setting; Williamson's theorem is the classification of positive-definite quadratic forms inside the symplectic group, parallel to the spectral theorem inside the orthogonal group.

• Symplectic group 05.01.03 — Williamson's theorem describes the orbits of $\mathrm{Sp}(2n,\mathbb{R})$ acting on real symmetric matrices by congruence; the symplectic spectrum is the complete orbit invariant in the positive-definite case.

• Symplectic manifold 05.01.02 — at any point of a symplectic manifold the linear symplectic structure on the tangent space lets one apply Williamson's theorem to the Hessian of a Hamiltonian at a critical point, defining the local frequencies of small oscillations.

• Birkhoff normal form 05.09.03 — Williamson is the quadratic-order linearisation; the Birkhoff iteration uses the symplectic eigenvalues as the frequency vector controlling resonance and convergence at higher orders.

• KAM theorem 05.09.01 — the Diophantine condition $|⟨k,\omega ⟩|\ge \gamma |k{|}^{-\tau }$ is imposed on the symplectic eigenvalues of the unperturbed Hamiltonian; these are exactly the Williamson invariants of its quadratic part on each invariant torus.

• Symplectic capacity / Gromov non-squeezing 05.07.01 — the symplectic capacity of an ellipsoid is $\pi$ divided by the largest symplectic eigenvalue of its defining matrix, so Williamson invariants are explicit obstructions to symplectic embedding.

• Action-angle coordinates 05.02.04 — for an integrable system, action-angle coordinates linearise the flow on each invariant torus to motion at the frequency vector $\omega (I)=\partial H_0/\partial I$; on a torus passing through an elliptic equilibrium, this frequency vector is the Williamson spectrum of the Hessian at the equilibrium.

• Hamiltonian vector field 05.02.01 — for a quadratic Hamiltonian $H_A$, the Hamiltonian vector field is $X_H=JA$, and Williamson's theorem is the conjugacy classification of these linear vector fields under the symplectic group.

The bridge from the algebraic input — a real symmetric positive-definite matrix — to each downstream geometric, dynamical, and quantum output runs through the symplectic spectrum. Putting these connections together, the same Williamson decomposition is the linear backbone of perturbation theory near elliptic equilibria, of the rigidity of symplectic embeddings of ellipsoids, and of the metaplectic quantisation of quadratic operators.

Historical & philosophical context [Master]

John Williamson's 1936 paper On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math. 58 (1936), 141-163 ^{[Williamson 1936]}, gave the complete classification of real symmetric matrices under congruence by the real symplectic group. The motivation came directly from the linearisation problem in classical mechanics: small oscillations near an equilibrium are governed by a quadratic Hamiltonian, and Williamson asked for the canonical forms to which any such Hamiltonian can be reduced by a symplectic change of variables. The 1936 paper handled both the positive-definite case (the simplest and most useful in mechanics) and the indefinite case, the latter requiring the four-block-type classification with elliptic-block signs.

The theorem entered the standard mechanics literature through Vladimir Arnold's Mathematical Methods of Classical Mechanics (1974, 2nd English edition 1989), where it appears as Appendix 6 ^[Arnold]. Arnold's exposition emphasised the role of the symplectic eigenvalues as the input data for Birkhoff normal-form theory and for the linearised KAM problem. Arnold-Kozlov-Neishtadt's Mathematical Aspects of Classical and Celestial Mechanics (1985, 3rd ed. 2006) ^{[Arnold-Kozlov-Neishtadt]} gave a fuller treatment with the Krein stability dichotomy and applications to celestial mechanics, including the parametric instability of triangular Lagrange points.

Yiming Long's 1971 work and subsequent monograph Index Theory for Symplectic Paths with Applications ^[Long] refined the classification at degenerate strata (multiple eigenvalues, nilpotent blocks at zero) and produced what is sometimes called the Williamson-Long form, central to the index theory of paths in $\mathrm{Sp}(2n,\mathbb{R})$ used in Conley-Zehnder and Maslov-index theory. Long's refinements are technically important for the count of periodic orbits in Hamiltonian systems and for the Morse-theoretic foundations of Floer homology.

Maurice de Gosson's Symplectic Geometry and Quantum Mechanics (2006) ^{[de Gosson]} developed the theorem's role in metaplectic quantisation and Gaussian quantum information. The Robertson-Schrödinger uncertainty principle, expressed as the lower bound ${\omega }_k\ge \hslash /2$ on each symplectic eigenvalue of a quantum covariance matrix, is the symplectic-invariant generalisation of Heisenberg's $\Delta q\Delta p\ge \hslash /2$ and is now the standard formulation in the Gaussian quantum-information literature.

Bibliography [Master]

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