Skew-Hamiltonian matrix explained

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

\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 matrixJ = \begin{bmatrix} 0 & I _{n} \\ - I _{n} & 0 \\ \end{bmatrix}

and I_n is the

n x n identity matrix.^[1] 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.^{[1] [2]}

Notes

1. [William C. Waterhouse]
2. [Heike Fassbender]