# 漢彌爾頓矩陣

${\displaystyle J={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}}}$

## 性質

${\displaystyle A={\begin{bmatrix}a&b\\c&d\end{bmatrix}}}$

## 漢彌爾頓算子

V為一個向量空間，在其上有著辛形式Ω。那麼當「${\displaystyle x,y\mapsto \Omega (A(x),y)}$是對稱的」這條件滿足時，就稱線性變換${\displaystyle A:\;V\mapsto V}$是一個對Ω漢彌爾頓算子（Hamiltonian operator），也就是說它當滿足下式：

${\displaystyle \Omega (A(x),y)=-\Omega (x,A(y))}$

## 參照

1. Ikramov, Khakim D., Hamiltonian square roots of skew-Hamiltonian matrices revisited, Linear Algebra and its Applications, 2001, 325: 101–107, doi:10.1016/S0024-3795(00)00304-9.
2. Meyer, K. R.; Hall, G. R., Introduction to Hamiltonian dynamical systems and the N-body problem, Springer, 1991, ISBN 0-387-97637-X.
3. ^ Dragt, Alex J., The symplectic group and classical mechanics, Annals of the New York Academy of Sciences, 2005, 1045 (1): 291–307, doi:10.1196/annals.1350.025.
4. Waterhouse, William C., The structure of alternating-Hamiltonian matrices, Linear Algebra and its Applications, 2005, 396: 385–390, doi:10.1016/j.laa.2004.10.003.
5. ^ Paige, Chris; Van Loan, Charles, A Schur decomposition for Hamiltonian matrices, Linear Algebra and its Applications, 1981, 41: 11–32, doi:10.1016/0024-3795(81)90086-0.