View source for Skew-Hamiltonian matrix

In [[Linear algebra|linear algebra]], skew-Hamiltonian matrices are special [[Matrix (mathematics)|matrices]] which correspond to [[Bilinear form#Symmetric.2C skew-symmetric and alternating forms|skew-symmetric bilinear forms]] on a [[Symplectic vector space|symplectic vector space]].
Let ''V'' be a [[Vector space|vector space]], equipped with a [[Symplectic vector space|symplectic form]] <math>\Omega</math>. Such a space must be even-dimensional. A linear map <math>A:\; V \mapsto V</math> is called '''a skew-Hamiltonian operator''' with respect to <math>\Omega</math> if the form  <math>x, y \mapsto \Omega(A(x), y)</math> is skew-symmetric.

Choose a basis <math> e_1, ... e_{2n}</math> in ''V'',  such that <math>\Omega</math> is written as <math>\sum_i e_i \wedge e_{n+i}</math>. Then a linear operator is skew-Hamiltonian with respect to <math>\Omega</math> if and only if its matrix ''A'' satisfies <math>A^T J = J A</math>, where ''J'' is the skew-symmetric matrix

:<math>J=
\begin{bmatrix}
0 & I_n \\
-I_n & 0 \\
\end{bmatrix}</math>

and ''I<sub>n</sub>'' is the <math>n\times n</math> [[Identity matrix|identity matrix]].<ref name=waterhouse>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</ref> 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.<ref name=waterhouse/><ref> 
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</ref>

==Notes==

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[[Category:Matrices]]
[[Category:Linear algebra]]



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