A symplectic vector space is a vector space $V$ over a field $F$ equipped with a symplectic bilinear form.

A symplectic blinear form is a mapping $\omega : V\times V\to F$ that is

• Bilinear: Linear is each argument separately
• Alternating 交替 : $\omega(v,v)=0,\forall v\in V$
• Non-degenerate 非退化 : $\omega(u,v)=0,\forall v\in V\Rightarrow u=0$
• 小结论：Blinear, Alternating ↔ skew-symmetry斜对称

Working in a fixed basis, $\omega$ can be represented by a matrix.

• skew-symmetric matrix
• nonsingular 非奇异 matrix | invertible| nondegenerate| regular matrix $\ker \omega =\{0\}\to AB=BA=I_n$

Invertible matrix| singular matrix 非|奇异矩阵

• hollow matrix which all diagonanl 对角线 entries are zero

In terms of basis vectors $(x_1,...,x_n,y_1,...,y_n)$

$$ \omega(u,v):=v^{T}\omega u\\\begin{aligned}\omega(x_i,y_j)&=-\omega(y_j,x_i)=\delta_{ij},\\\omega(x_i,x_j)&=\omega(y_i,y_j)=0.\end{aligned} $$

The standard sympletic space is $\mathbb{R} ^{2n}$ with the sympletic form given by a nonsingular, skew-symmetric matrix. Typically $\omega$ is chosen to be the block matrix

$$ \omega=\begin{bmatrix}0&I_n\\-I_n&0\end{bmatrix} $$

A modified version of the Gram-Schmidt show that any finite-dimensional sympletic vector space has a basis such that $\omega$ take this form, often called a Darboux basis| sympletic basis.

Lagrangian form