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

Given a basis  $e\_1,\; \backslash ldots,\; e\_\{2n\}$  in  $V$ , the symplectic form  Ω  can be expressed as  $\backslash sum\_\{i\}\; e\_i\; \backslash wedge\; e\_\{n+i\}$ . In this context, a linear operator $A$ is skew-Hamiltonian with respect to Ω if and only if its corresponding matrix satisfies the condition  $A^T\; J\; =\; J\; A$, 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  $n\; \backslash times\; n$  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.William C. Waterhouse, [http://linkinghub.elsevier.com/retrieve/pii/S0024379504004410 The structure of alternating-Hamiltonian matrices], Linear Algebra and its Applications, Volume 396, 1 February 2005, Pages 385-390

Heike Fassbender, D. Steven Mackey, Niloufer Mackey and Hongguo Xu

[http://www.icm.tu-bs.de/~hfassben/papers/hamsqrt.pdf Hamiltonian Square Roots of Skew-Hamiltonian Matrices],

Linear Algebra and its Applications 287, pp. 125 - 159, 1999

Category:Matrices (mathematics)

Category:Linear algebra

{{matrix-stub}}