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

J= \begin{bmatrix} 0 & I_n \\ -I_n & 0 \\ \end{bmatrix}

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.

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