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 $$\Omega$$. Such a space must be even-dimensional. A linear map $$A:\; V \mapsto V$$ is called a skew-Hamiltonian operator with respect to $$\Omega$$ if the form $$x, y \mapsto \Omega(A(x), y)$$ is skew-symmetric.

Choose a basis $$ e_1, ... e_{2n}$$ in V, such that $$\Omega$$ is written as $$\sum_i e_i \wedge e_{n+i}$$. Then a linear operator is skew-Hamiltonian with respect to $$\Omega$$ if and only if its matrix A satisfies $$A^T J = J A$$, where J is the skew-symmetric matrix


 * $$J=

\begin{bmatrix} 0 & I_n \\ -I_n & 0 \\ \end{bmatrix}$$

and In is the $$n\times n$$ 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.