Symplectic matrix

An invertible real matrix preserving the standard symplectic form. With $J=\left(\begin{array}{cc}0& I\\ -I& 0\end{array}\right)$, it satisfies $A^{T}JA=J$, equivalently $AJ{A}^T=J$. The first identity describes preservation of the form; the second describes preservation of coordinate Poisson brackets.

A **symplectic matrix** is a special type of square matrix that preserves a symplectic form. Symplectic matrices are used primarily in the context of symplectic geometry and Hamiltonian mechanics.