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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 03:35, 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 . Such a space must be even-dimensional. A linear map is called a skew-Hamiltonian operator with respect to if the form is skew-symmetric.
Choose a basis in V, such that is written as . Then a linear operator is skew-Hamiltonian with respect to if and only if its matrix A satisfies , where J is the skew-symmetric matrix
and In is the 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
- ↑ 1.0 1.1 William C. Waterhouse, The structure of alternating-Hamiltonian matrices, Linear Algebra and its Applications, Volume 396, 1 February 2005, Pages 385-390
- ↑ 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