Symplectic matrix

In mathematics, a symplectic matrix is a ${\displaystyle 2n\times 2n}$ matrix ${\displaystyle M}$ with real entries that satisfies the condition

${\displaystyle M^{\text{T}}\Omega M=\Omega ,}$

(1)

where ${\displaystyle M^{\text{T}}}$ denotes the transpose of ${\displaystyle M}$ and ${\displaystyle \Omega }$ is a fixed ${\displaystyle 2n\times 2n}$ nonsingular, skew-symmetric matrix. This definition can be extended to ${\displaystyle 2n\times 2n}$ matrices with entries in other fields, such as the complex numbers, finite fields, p-adic numbers, and function fields.

Typically ${\displaystyle \Omega }$ is chosen to be the block matrix $${\displaystyle \Omega ={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}},}$$ where ${\displaystyle I_{n}}$ is the ${\displaystyle n\times n}$ identity matrix. The matrix ${\displaystyle \Omega }$ has determinant ${\displaystyle +1}$ and its inverse is ${\displaystyle \Omega ^{-1}=\Omega ^{\text{T}}=-\Omega }$.