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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]].
 
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.
 
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]].<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==
 
<references />
 
[[Category:Matrices]]
[[Category:Linear algebra]]
 
 
{{Linear-algebra-stub}}

Revision as of 02:35, 17 March 2013

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

Choose a basis e1,...e2n in V, such that Ω is written as ∑iei∧en+i. Then a linear operator is skew-Hamiltonian with respect to Ω if and only if its matrix A satisfies ATJ=JA, where J is the skew-symmetric matrix

J=[0In−In0]

and In is the n×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

  1. ↑ 1.0 1.1 William C. Waterhouse, The structure of alternating-Hamiltonian matrices, Linear Algebra and its Applications, Volume 396, 1 February 2005, Pages 385-390
  2. ↑ 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


Template:Linear-algebra-stub

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