Definition:Symplectic Basis

Definition

Let $\mathbb K$ be a field.

Let $\struct {V, f}$ be a bilinear space over $\mathbb K$ of finite dimension $2 n > 0$.

Let $\BB = \tuple {b_1, c_1, \ldots, b_n, c_n}$ be an ordered basis of $V$.

Then $\BB$ is symplectic if and only if:

$\map f {b_i, b_j} = \map f {c_i, c_j} = 0$ for all $i, j$

$\map f {b_i, c_j} = \delta_{i j}$ for all $i, j$

where $\delta$ denotes Kronecker delta.

That is, if and only if the matrix of $f$ relative to $\BB$ has the form:

$\begin{pmatrix}

0 & 1 \\

-1 & 0 \\

& &  0 & 1 \\
& & -1 & 0 \\
& & & & \ddots \\
& & & & &  0 & 1 \\ 
& & & & & -1 & 0

\end{pmatrix}$

Weisstein, Eric W. "Symmetric Bilinear Form." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SymmetricBilinearForm.html