# Definition A *symplectic form* $\Omega$ on a [[vector space]] $Z$ is a [[Non-Degeneracy (Quadratic Forms)|non-degenerate]] [[Bilinear Form|bilinear form]] on $Z$. # Example: Symplectic Form Associated to an Inner Product Space Let $(W, \braket{\cdot|\cdot})$ be a real [[inner product space]], since $W$ is in [[Dual Space|duality]] with itself, then we obtain a symplectic form on $Z = W \times W$: $ \Omega((w_1, v_1),(w_2, v_2)) = \braket{v_2|w_1} - \braket{v_1|w_2} $ $v_1, v_2, w_1, w_2 \in W$ In the case of $W = \mathbb{R}^3$, we have $\Omega: \mathbb{R}^6 \rightarrow \mathbb{R}$ $ \Omega((\pmb{w}_1, \pmb{v}_1),(\pmb{w}_2, \pmb{v}_2)) = \pmb{v}_2\cdot \pmb{w}_1 - \pmb{v}_1 \cdot \pmb{w}_2 $ where $\cdot$ is the usual [[inner product]] on $\mathbb{R}^3$, and we have identified $\mathbb{R}^3$ with $(\mathbb{R}^3)^*$. We can bring $\Omega$ to "canonical form" using standard linear algebra. In this case, it will have 36 [[components]]. The [[basis]] vectors are: $ \begin{align} f_1 \equiv (\pmb{e}_1, \pmb{e}_1) \\ f_2 \equiv (\pmb{e}_1, \pmb{e}_2) \\ f_3 \equiv (\pmb{e}_1, \pmb{e}_3)\\ f_4 \equiv (\pmb{e}_2, \pmb{e}_2)\\ f_5 \equiv (\pmb{e}_2, \pmb{e}_3)\\ f_6 \equiv (\pmb{e}_3, \pmb{e}_3) \end{align} $ $ [\Omega]_{IJ}= \Omega(f_I = (\pmb{e}_i, \pmb{e}_j),f_J = (\pmb{e}_k, \pmb{e}_l)) = \delta_{li} - \delta_{kj} $ This can be evaluated to: $ [\Omega]_{IJ} = \mathbb{J} $ where $\mathbb{J}$ is the [[Symplectic Matrix]] with $n=6$. This generalizes to more abstract finite-dimensional spaces. We can conclude: 1. [[Symplectic Vector Space|Symplectic vector spaces]] are even dimensional. 2. There is always a basis for such spaces in which the matrix for $\Omega$ is actually $\mathbb{J}$. Such a basis is called *canonical*, as are the corresponding coordinates.