Hamiltonian Matrix - Hamiltonian Operators

Hamiltonian Operators

Let V be a vector space, equipped with a symplectic form Ω. A linear map is called a Hamiltonian operator with respect to Ω if the form is symmetric. Equivalently, it should satisfy

Choose a basis e₁, ... e_2n in V, such that Ω is written as . A linear operator is Hamiltonian with respect to Ω if and only if its matrix in this basis is Hamiltonian.

From this definition, the following properties are apparent. A square of a Hamiltonian matrix is skew-Hamiltonian. An exponential of a Hamiltonian matrix is symplectic, and a logarithm of a symplectic matrix is Hamiltonian.