Hamiltonian matrix

Suppose that the -by- matrix is written as the block matrix : A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} where , , , and are -by- matrices. Then the condition that be Hamiltonian is equivalent to requiring that the matrices and are symmetric, and that . Another equivalent condition is that is of the form with symmetric. The characteristic polynomial of a real Hamiltonian matrix is even. Thus, if a Hamiltonian matrix has as an eigenvalue, then , and are also eigenvalues. ==Extension to complex matrices==

As for symplectic matrices, the definition for Hamiltonian matrices can be extended to complex matrices in two ways. One possibility is to say that a matrix is Hamiltonian if , as above. ==Hamiltonian operators==

Let be a vector space, equipped with a symplectic form . A linear map A : \; V \mapsto V is called a Hamiltonian operator with respect to if the form x, y \mapsto \Omega(A(x), y) is symmetric. Equivalently, it should satisfy :\Omega(A(x), y) = -\Omega(x, A(y)) Choose a basis in , such that is written as \sum_i e_i \wedge e_{n+i}. A linear operator is Hamiltonian with respect to if and only if its matrix in this basis is Hamiltonian. ==References==