Hamiltonian matrix

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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

$K= \begin{bmatrix} 0 & I_n \\ -I_n & 0 \\ \end{bmatrix}$

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

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

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

1 Properties
2 Hamiltonian operators
3 See also
4 References
5 Notes

[edit] Properties

Let $M$ be a 2n×2n block matrix given by

$M = \begin{pmatrix}A & B \\ C & D\end{pmatrix}$

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 ${\mathfrak{Sp}}(2n)$ .^[1]

[edit] Hamiltonian operators

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

Ω(A (x), y) = - Ω(x, A (y))

Choose a basis $e 1,... e 2 n$ in V, such that $Ω$ is written as $\sum_i e_i \wedge e_{n+i}$ . A linear operator is Hamiltonian with respect to $Ω$ if and only if its matrix in this basis is Hamiltonian.^[2]

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.

[edit] See also

Symplectic matrix

[edit] References

K.R.Meyer, G.R. Hall (1991). Introduction to Hamiltonian dynamical systems and the 'N'-body problem. Springer, pp. 34-35. ISBN 0-387-97637-X.

[edit] Notes

^ Alex J. Dragt, The Symplectic Group and Classical Mechanics'' Annals of the New York Academy of Sciences (2005) 1045 (1), 291-307.
^ William C. Waterhouse, The structure of alternating-Hamiltonian matrices, Linear Algebra and its Applications, Volume 396, 1 February 2005, Pages 385-390