In linear algebra, a skew-Hamiltonian matrix is a specific type of matrix that corresponds to a skew-symmetric bilinear form on a symplectic vector space.
be a vector space equipped with a symplectic form, denoted by Ω.
A symplectic vector space must necessarily be of even dimension.
is defined as a skew-Hamiltonian operator with respect to the symplectic form Ω if the bilinear form defined by
, the symplectic form Ω can be expressed as
In this context, a linear operator
is skew-Hamiltonian with respect to Ω if and only if its corresponding matrix satisfies the condition
is the skew-symmetric matrix defined as: With
Notably, the square of any Hamiltonian matrix is skew-Hamiltonian.
This article about matrices is a stub.