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. Such a space must be even-dimensional. A linear map is called a skew-Hamiltonian operator with respect to if the form is skew-symmetric.
Choose a basis in V, such that is written as. Then a linear operator is skew-Hamiltonian with respect to if and only if its matrix A satisfies, where J is the skew-symmetric matrix
and In is the 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.
