Symplectic matrix

A symplectic matrix $\mathrm{M}$ is a $2n\times2n$ matrix which satisfies the following condition:

where $^{T}$ denotes transposition and $\mathrm{E}$ is some $2n\times 2n$ antisymmetric invertible matrix.

Typically, $\mathrm{E}$ is chosen to be the block matrix

where $\mathrm{I}_{n}$ is the $n$-dimensional Identity matrix. This is known as the ($2n$-dimensional) standard symplectic matrix. In the simplest possible form as a $2\times 2$ matrix, it is

It appears frequently in areas that deal with rotations and symplectic geometry. For example, it is the generating matrix of the special orthogonal group $SO(2)$. It is also ubiquitous in the treatment of the Hamilton equations of motions.

Properties#

• Symplectic matrices form a group.
• $\mathrm{E}$ has unit determinant: $\det \mathrm{E}=1$.
• The identity matrix is symplectic, since $\mathrm{I}\mathrm{E}\mathrm{I}=\mathrm{E}$.