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 \( A^{T}J=JA \) , where J is the skew-symmetric matrix
\( J={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}} \)
and In is the \( n\times n \) identity matrix.[1] 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.[1][2]
Notes
William C. Waterhouse, 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 Hamiltonian Square Roots of Skew-Hamiltonian Matrices, Linear Algebra and its Applications 287, pp. 125 - 159, 1999
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