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

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

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

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

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

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

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 eigenvalues of $M$ are symmetric about the imaginary axis.

The space of all Hamiltonian matrices is a Lie algebra ${\mathfrak {Sp}}(2n)$.

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, it should 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.

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.

See also

• Symplectic matrix

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.