Hamiltonian matrix

In mathematics, a Hamiltonian matrix "A" is any real "2n×2n" matrix that satisfies the condition that "KA" is symmetric, where "K" is the skew-symmetric matrix

:

and "I_n" is the "n×n" identity matrix. In other words, $A$ is Hamiltonian if and only if

:$KA\; -\; A^T\; K^T\; =\; KA\; +\; A^T\; K\; =\; 0.,$

In the vector space of all "2n×2n" matrices, Hamiltonian matrices form a "2n² + n" vector subspace.

Properties

* Let $M$ be a "2n×2n" block matrix given by:where $A,\; B,\; C,\; D$ are "n×n" matrices. Then $M$ is a Hamiltonian matrix provided that matrices $B,\; C$ are symmetric, and $A\; +\; D^T\; =\; 0$.
* The transpose of a Hamiltonian matrix is Hamiltonian.
* The trace of a Hamiltonian matrix is zero.
* Commutator of two Hamiltonian matrices is Hamiltonian.The space of all Hamiltonian matrices is a Lie algebra $\{mathfrak\{Sp(2n)$. [Alex J. Dragt, [http://www.blackwell-synergy.com/doi/abs/10.1196/annals.1350.025 "The Symplectic Group and Classical Mechanics'] ' Annals of the New York Academy of Sciences (2005) 1045 (1), 291-307. ]

Hamiltonian operators

Let "V" be a vector space, equipped with a symplectic form $Omega$. A linear map $A:;\; V\; mapsto\; V$ is called a Hamiltonian operator with respect to $Omega$ if the form $x,\; y\; mapsto\; Omega(A(x),\; y)$ is symmetric. Equivalently, itshould satisfy

:$Omega(A(x),\; y)=-Omega(x,\; A(y))$

Choose a basis $e\_1,\; ...\; e\_\{2n\}$ in "V", such that $Omega$ is written as $sum\_i\; e\_i\; wedge\; e\_\{n+i\}$. A linear operator is Hamiltonian with respect to $Omega$ if and only if its matrix in this basis is Hamiltonian. [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]

From this definition, the following properties are apparent.A square of a Hamiltonian matrix is skew-Hamiltonian. An exponential of a Hamiltonian matrix is symplectic, and a logarithm of a symplectic matrix is Hamiltonian.

ee also

*Symplectic matrix

References

* cite book | author=K.R.Meyer, G.R. Hall | title=Introduction to Hamiltonian dynamical systems and the 'N'-body problem | publisher = Springer
year = 1991
pages = pp. 34-35
id = ISBN 0-387-97637-X

Notes