Hamiltonian matrix

In mathematics, a Hamiltonian matrix is a $2 n$ -by- $2 n$ matrix $A$ such that $JA$ is symmetric, where $J$ is the skew-symmetric matrix $J={\begin{bmatrix}0_{n}&I_{n}\\-I_{n}&0_{n}\end{bmatrix}}$ and $I n$ is the $n$ -by- $n$ identity matrix. In other words, $A$ is Hamiltonian if and only if $(JA) = JA$ where $( )$ denotes the transpose. The collection of all Hamiltonian matrices forms a Lie algebra (the symplectic Lie algebra); its associated Lie group is the symplectic group, whose elements are the symplectic matrices.