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| In [[linear algebra]], skew-Hamiltonian matrices are special [[Matrix (mathematics)|matrices]] which correspond to [[skew-symmetric]] [[bilinear form]]s on a [[symplectic vector space]].
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| Let ''V'' be a [[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.
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| 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
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| :<math>J=
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| \begin{bmatrix}
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| 0 & I_n \\
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| -I_n & 0 \\
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| \end{bmatrix}</math>
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| and ''I<sub>n</sub>'' is the <math>n\times n</math> [[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'''. | |
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| 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>
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| Heike Faßbender, D. Steven Mackey, Niloufer Mackey and Hongguo Xu
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| [http://www.icm.tu-bs.de/~hfassben/papers/hamsqrt.pdf Hamiltonian Square Roots of Skew-Hamiltonian Matrices,]
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| Linear Algebra and its Applications 287, pp. 125 - 159, 1999</ref>
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| ==Notes==
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| <references />
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| [[Category:Matrices]]
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| [[Category:Linear algebra]]
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| {{Linear-algebra-stub}}
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Latest revision as of 20:13, 7 October 2014
The author's name is Andera and she thinks it sounds fairly great. What me and my family love psychics is bungee leaping but I've been taking on new issues recently. North Carolina is the location he loves most but now he is considering other choices. He is an info officer.