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Write vectors in a symplectic basis as $v={\sum }_i(q_i{e}_i+p_i{f}_i)$ and $w={\sum }_i(Q_i{e}_i+P_i{f}_i)$. Their coordinate vectors and the matrix of a bilinear form areHere $I_n$ is the identity matrix, and $T$ denotes transpose. The ordering matters: interleaving the basis as $(e_1,f_1,\dots ,e_n,f_n)$ instead gives a block diagonal matrix with $2\times 2$ blocks $\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$. Some conventions reverse every $f_i$, giving $-J$ instead. Always check the displayed pairing convention.^[ref]

Over the real numbers, each summand is the signed area of a parallelogram in the $(q_i,p_i)$ coordinate plane. For $a=(2,1,1,1{)}^T$ and $b=(1,1,0,2{)}^T$, the two signed areas are $-1$ and $1$, so $\omega (a,b)=0$ although the vectors are linearly independent. The cancellation concerns the symplectic form; it does not assert that their dot product vanishes. Over a general field, the same determinant formula is algebraic, without a signed geometric interpretation.

A symplectic basis organizes a nondegenerate alternating bilinear form into independent pairs. Let $(V,\omega )$ be a finite-dimensional vector space over a field $K$, with $\omega$ a nondegenerate bilinear form satisfying $\omega (v,v)=0$ for every vector $v$. Its dimension is even, say $2n$, and a symplectic basis is an ordered basis $\mathcal{B}=(e_1,\dots ,e_n,f_1,\dots ,f_n)$ such thatHere ${\delta }_{ij}$ is the Kronecker delta. Each $(e_i,f_i)$ is a hyperbolic pair; different pairs have zero mutual pairing. These conventions apply in every characteristic. In characteristic two, $-1=1$, but the condition $\omega (v,v)=0$ is still essential: a symmetric bilinear form need not be an alternating bilinear form. The existence theorem and the characteristic caveat are given in Bilinear Forms by Keith Conrad (Keith Conrad, Bilinear Forms, §1, Theorem 1.6; §5, Theorems 5.1 and 5.4 and Definition 5.3).

The benefit is a common coordinate language for every symplectic vector space of the same finite dimension over the same field. The span of all $e_i$, and separately of all $f_i$, is a Lagrangian subspace; each paired span is a symplectic subspace. In Hamiltonian mechanics, the corresponding pairing of position and momentum coordinates describes phase space and canonical transformations. On a symplectic manifold, a symplectic basis describes one tangent space; Darboux coordinates additionally describe a whole local neighborhood.^[ref]