Skew Hamiltonian matrix

In linear algebra, skew-Hamiltonian matrices are special matrices which correspond to skew-symmetric bilinear forms on a symplectic vector space.

Let V be a vector space, equipped with a symplectic form Ω . Such a space must be even-dimensional. A linear map A : V ↦ V is called a skew-Hamiltonian operator with respect to Ω if the form x , y ↦ Ω ( A ( x ) , y ) is skew-symmetric.

Choose a basis e 1 , . . . e 2 n in V, such that Ω is written as ∑ i e i ∧ e n + i . Then a linear operator is skew-Hamiltonian with respect to Ω if and only if its matrix A satisfies A T J = J A , where J is the skew-symmetric matrix

and I_n is the n × n identity matrix. Such matrices are called skew-Hamiltonian.

The square of a Hamiltonian matrix is skew-Hamiltonian. The converse is also true: every skew-Hamiltonian matrix can be obtained as the square of a Hamiltonian matrix.