Then the condition that A be Hamiltonian is equivalent to requiring that the matrices b and c are symmetric, and that a + dT = 0.
[1][2] Another equivalent condition is that A is of the form A = JS with S symmetric.
It follows that the space of all Hamiltonian matrices is a Lie algebra, denoted sp(2n).
However the logarithm of a symplectic matrix is not necessarily Hamiltonian because the exponential map from the Lie algebra to the group is not surjective.
[2]: 34–36 [3] The characteristic polynomial of a real Hamiltonian matrix is even.
[5] Let V be a vector space, equipped with a symplectic form Ω.
is called a Hamiltonian operator with respect to Ω if the form
Equivalently, it should satisfy Choose a basis e1, …, e2n in V, such that Ω is written as