Symplectic form (awaiting review)

A symplectic form on a vector space $V$ over a field $k$ is a non-degenerate alternating bilinear form $\omega\colon V\times V\to k$. This means that

• if $\omega(u,v)=0$ for all $v\in V$ then $u=0$ (non-degenerate);
• $\omega(v,v)=0$ for all $v\in V$ (alternating);
• $\omega(\lambda u+v,w)=\lambda\omega(u,v)+\omega(v,w)$ and $\omega(u,\lambda v+w)=\omega(u)+\lambda\omega(v,w)$ for all $\lambda\in k$, $u,v,w\in V$ (bilinear).

A finite dimensional vector space admitting a symplectic form $\omega$ necessarily has even dimension $2n$, and in this case $\omega$ can be represented by a matrix $\Omega\in k^{2n\times 2n}$ that satisfies $u^\intercal\Omega v=\omega(u,w)$ for all $u,v\in V$. One can always choose a basis for $V$ so that $\Omega = \begin{bmatrix} 0 & I_n\\ -I_n & 0\end{bmatrix},$ where $I_n$ denotes the $n\times n$ identity matrix.