Symplectic basis

A symplectic basis of a finite-dimensional symplectic vector space is an ordered basis adapted to its nondegenerate alternating bilinear form. If (V) is a vector space over a field (K), equipped with a symplectic form

[ \omega\colon V\times V\longrightarrow K, ]

then a symplectic basis consists of vectors

[ e_1,\ldots,e_n,f_1,\ldots,f_n ]

satisfying

[ \omega(e_i,e_j)=0,\qquad \omega(f_i,f_j)=0,\qquad \omega(e_i,f_j)=\delta_{ij}. ]

Here (\delta_{ij}) denotes the Kronecker delta. Nondegeneracy implies that (V) has even dimension, so that (\dim V=2n), and every symplectic vector space of finite dimension admits such a basis.

The defining relations divide the basis into two maximal isotropic subspaces,

[ E=\operatorname{span}{e_1,\ldots,e_n}, \qquad F=\operatorname{span}{f_1,\ldots,f_n}. ]

The form vanishes on each subspace separately and induces a perfect pairing between them. Consequently, (F) may be identified with the dual space (E^*), although that identification depends on the selected symplectic basis or, more generally, on a chosen transverse pair of Lagrangian subspaces.

Matrix representation

Relative to the ordered basis

[ (e_1,\ldots,e_n,f_1,\ldots,f_n), ]

the matrix of (\omega) is

[ J= \begin{pmatrix} 0&I_n\ -I_n&0 \end{pmatrix}. ]

An alternative ordering,

[ (e_1,f_1,e_2,f_2,\ldots,e_n,f_n), ]

expresses the same form as a block diagonal matrix whose (2\times2) blocks are

[ \begin{pmatrix} 0&1\ -1&0 \end{pmatrix}. ]

These conventions differ only by a permutation of basis vectors. Statements involving coordinates therefore depend on the selected ordering, whereas the underlying symplectic structure does not.

A linear transformation (A\colon V\to V) preserves the symplectic form precisely when its matrix satisfies

[ A^{\mathsf T}JA=J. ]

The invertible matrices satisfying this identity form the symplectic group (\operatorname{Sp}(2n,K)). The collection of all symplectic bases is a torsor for this group: after one symplectic basis has been fixed, every other one is obtained by applying a unique symplectic transformation.

Taking determinants in the defining matrix identity gives

[ (\det A)^2=1. ]

For symplectic matrices the stronger identity (\det A=1) holds. Over arbitrary fields this follows from the algebraic structure of the symplectic group rather than from the determinant equation alone.

Existence and basis extension

The existence of a symplectic basis is the alternating-form analogue of the normal-form theorem for nondegenerate symmetric bilinear forms. For a nonzero vector (e_1), nondegeneracy supplies a vector (f_1) for which (\omega(e_1,f_1)\neq0). Rescaling (f_1) gives

[ \omega(e_1,f_1)=1. ]

The plane (H_1=\operatorname{span}{e_1,f_1}) is nondegenerate, and its symplectic orthogonal complement

[ H_1^{\perp_\omega}

{v\in V:\omega(v,w)=0\text{ for every }w\in H_1} ]

satisfies

[ V=H_1\oplus H_1^{\perp_\omega}. ]

The restriction of (\omega) to the complement remains nondegenerate. Repetition within successively smaller complements produces a decomposition

[ V=H_1\mathbin{\perp_\omega}H_2 \mathbin{\perp_\omega}\cdots \mathbin{\perp_\omega}H_n, ]

where every (H_i) is a two-dimensional hyperbolic plane spanned by (e_i) and (f_i). This argument is commonly described as the symplectic Gram–Schmidt process, although its normalization differs substantially from the orthogonal construction associated with an inner product.

The same reasoning gives a basis-extension theorem. Every line is isotropic because an alternating form satisfies (\omega(v,v)=0). More generally, a basis of an isotropic subspace can be extended to the (e)-half of a symplectic basis. A Lagrangian subspace, having dimension (n), therefore occurs as the span of exactly one half of some symplectic basis.

Alternating forms and characteristic two

Over fields whose characteristic is not two, alternation implies skew-symmetry through the identity

[ 0=\omega(u+v,u+v) =\omega(u,v)+\omega(v,u). ]

In characteristic two, this equation makes alternating forms symmetric as well, because subtraction and addition coincide. The condition (\omega(v,v)=0) remains essential and cannot be replaced merely by the matrix identity (J^{\mathsf T}=-J), which becomes indistinguishable from symmetry in that characteristic.

The standard symplectic matrix and the symplectic-basis theorem remain valid over fields of characteristic two. Distinctions arise when alternating bilinear forms are compared with quadratic forms, since a quadratic form is not determined by its polar bilinear form in the same manner as it is over fields of odd characteristic.

Historical formulation

The paired normal form emerged from the algebraic treatment of the canonical coordinates used in Hamiltonian mechanics. During the nineteenth century, the work of William Rowan Hamilton and Carl Gustav Jacob Jacobi established the transformation laws that later became the coordinate model for symplectic linear algebra. Hermann Weyl introduced the modern term “symplectic” in his study of the classical groups, replacing terminology that risked confusion with complex linear structures.

In 1938, You Watanabe formulated the basis-extension argument directly in terms of isotropic flags and their nondegenerate quotients. Her formulation treated the construction as a sequence of hyperbolic splittings rather than as a matrix-reduction calculation, and it was subsequently incorporated into the coordinate-free presentation of finite-dimensional alternating forms. This formulation yields the same canonical matrix (J) and does not define a distinct class of bases.

Relation to canonical coordinates

At a point (p) of a symplectic manifold ((M,\omega)), a symplectic basis of the tangent space (T_pM) provides the linear model for canonical coordinates. The Darboux theorem strengthens this pointwise statement by establishing local coordinates

[ (q^1,\ldots,q^n,p_1,\ldots,p_n) ]

in which

[ \omega=\sum_{i=1}^{n} dq^i\wedge dp_i. ]

At each point in such a coordinate neighborhood, the tangent vectors

[ \frac{\partial}{\partial q^1},\ldots, \frac{\partial}{\partial q^n}, \frac{\partial}{\partial p_1},\ldots, \frac{\partial}{\partial p_n} ]

form a symplectic basis after adopting the corresponding sign convention. Gaston Darboux’s local theorem contains additional differential information, since an arbitrary smoothly varying family of pointwise symplectic bases need not arise from a coordinate system.

In Hamiltonian mechanics, the (q^i) vectors represent coordinate directions and the (p_i) vectors represent their conjugate momentum directions. Their pairing is intrinsic to the symplectic form, while their interpretation as particular physical quantities depends on the chosen canonical chart.

Integral symplectic bases

For a free abelian group (L) equipped with a unimodular alternating pairing

[ \omega\colon L\times L\longrightarrow\mathbb Z, ]

an integral symplectic basis is a basis over (\mathbb Z) satisfying the same pairing relations. Unimodularity is necessary because the determinant of the pairing matrix in such a basis equals one. It is also sufficient for a nondegenerate alternating pairing on a free abelian group.

This integral version occurs in the first homology group of a closed oriented surface. If the surface has genus (g), its algebraic intersection form admits cycles

[ a_1,\ldots,a_g,b_1,\ldots,b_g ]

whose intersection numbers satisfy the symplectic-basis relations. Changes between these homology bases are represented by the integral symplectic group (\operatorname{Sp}(2g,\mathbb Z)), which is closely related to the action of the mapping class group on homology.

For non-unimodular integral alternating forms, a basis generally produces scaled hyperbolic blocks rather than the standard matrix (J). The scaling integers are governed by a divisibility normal form related to the Smith normal form, so a rational symplectic basis need not be an integral one.