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 $A^{T}J=JA$, where J is the skew-symmetric matrix

$J={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}}$

and I_n 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.