Hamiltonian matrix
From Wikipedia, the free encyclopedia
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 In is the n×n identity matrix. In other words, A is Hamiltonian if and only if
In the vector space of all 2n×2n matrices, Hamiltonian matrices form a 2n2 + n vector subspace.
[edit] 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 + DT = 0.
- The transpose of a Hamiltonian matrix is Hamiltonian.
- The trace of a Hamiltonian matrix is zero.
- The product of two Hamiltonian matrices is Hamiltonian if and only if the matrices commute.
Properties 2 and 3 follow from the first property.
[edit] See also
[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.
 |
This mathematics-related article is a stub. You can help Wikipedia by expanding it. |
