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

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

[math]\displaystyle{ J= \begin{bmatrix} 0 & I_n \\ -I_n & 0 \\ \end{bmatrix} }[/math]

and I_n is the [math]\displaystyle{ n\times n }[/math] 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.