Skew-Hamiltonian matrix

Skew-Hamiltonian Matrices in Linear Algebra

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

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

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

With  $I_{n}$  representing the  $n\times n$  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.