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 formula_1. Such a space must be even-dimensional. A linear map formula_2 is called a skew-Hamiltonian operator with respect to formula_1 if the form formula_4 is skew-symmetric.
Choose a basis formula_5 in "V", such that formula_1 is written as formula_7. Then a linear operator is skew-Hamiltonian with respect to formula_1 if and only if its matrix "A" satisfies formula_9, where "J" is the skew-symmetric matrix
and "In" is the formula_11 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.