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 $Omega$. Such a space must be even-dimensional. A linear map $A:;\; V\; mapsto\; V$ is called a skew-Hamiltonian operator with respect to $Omega$ if the form $x,\; y\; mapsto\; Omega(A(x),\; y)$ is skew-symmetric.

Choose a basis $e\_1,\; ...\; e\_\{2n\}$ in "V", such that $Omega$ is written as $sum\_i\; e\_i\; wedge\; e\_\{n+i\}$. Then a linear operator is skew-Hamiltonian with respect to $Omega$ if and only if its matrix "A" satisfies $A^T\; J\; =\; J\; A$, where "J" is the skew-symmetric matric

:

and "I_n" is the $n\; imes\; n$ identity 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] 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.William C. Waterhouse, "The structure of alternating-Hamiltonian matrices", Linear Algebra and its Applications, Volume 396, 1 February 2005, Pages 385-390]

Notes