Skew-Hamiltonian matrix

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. Let ${\displaystyle V}$be a vector space equipped with a symplectic form, denoted by Ω. A symplectic vector space must necessarily be of even dimension.

A linear map ${\displaystyle A:\;V\mapsto V}$is defined as a skew-Hamiltonian operator with respect to the symplectic form Ω if the bilinear form defined by ${\displaystyle (x,y)\mapsto \Omega (A(x),y)}$ is skew-symmetric.

Given a basis  ${\displaystyle e_{1},\ldots ,e_{2n}}$  in  ${\displaystyle V}$, the symplectic form  Ω  can be expressed as  ${\textstyle \sum _{i}e_{i}\wedge e_{n+i}}$. In this context, a linear operator ${\displaystyle A}$is skew-Hamiltonian with respect to Ω if and only if its corresponding matrix satisfies the condition  ${\displaystyle A^{T}J=JA}$, where  ${\displaystyle J}$  is the skew-symmetric matrix defined as:

${\displaystyle J={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}}}$

With  ${\displaystyle I_{n}}$  representing the  ${\displaystyle n\times n}$  identity matrix.

Matrices that meet this criterion are classified as skew-Hamiltonian matrices. Notably, the square of any Hamiltonian matrix is skew-Hamiltonian. Conversely, any skew-Hamiltonian matrix can be expressed as the square of a Hamiltonian matrix.