Definition:Symplectic Basis

Definition
Let $\mathbb K$ be a field.

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

Let $\mathcal B = \left({b_1, c_1, \ldots, b_n, c_n}\right)$ be an ordered basis of $V$.

Then $\mathcal B$ is symplectic :
 * $f \left({b_i, b_j}\right) = f \left({c_i, c_j}\right) = 0$ for all $i, j$
 * $f \left({b_i, c_j}\right) = \delta_{ij}$ for all $i, j$

where $\delta$ denotes Kronecker delta.

That is, the matrix of $f$ relative to $\mathcal B$ has the form:
 * $\begin{pmatrix}

0 & 1 \\ -1 & 0 \\ & & 0 & 1 \\ & & -1 & 0 \\ & & & & \ddots \\ & & & & & 0 & 1 \\  & & & & & -1 & 0 \end{pmatrix}$

Also see

 * Definition:Orthogonal Basis (Bilinear Space)