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

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 $\{mathfrak\{Sp\}\}(2n)$.

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. ISBN 0-387-97637-X.

Notes