Hamiltonian matrix

Template:Short descriptionTemplate:Use dmy dates In mathematics, a Hamiltonian matrix is a 2n-by-2n matrix Template:Mvar such that JA is symmetric, where Template:Mvar is the skew-symmetric matrix $$J=\left[\begin{array}{cc}{0}_n& I_n\\ -I_n& 0_n\end{array}\right]$$ and I_n is the Template:Mvar-by-Template:Mvar identity matrix. In other words, Template:Mvar is Hamiltonian if and only if (JA)^T = JA where ( )^T denotes the transpose. The collection of all Hamiltonian matrices forms a Lie algebra (the symplectic Lie algebra); its associated Lie group is the symplectic group, whose elements are the symplectic matrices.

Properties

Suppose that the 2n-by-2n matrix Template:Mvar is written as the block matrix

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

where Template:Mvar, Template:Mvar, Template:Mvar, and Template:Mvar are Template:Mvar-by-Template:Mvar matrices. Then the condition that A be Hamiltonian is equivalent to requiring that the matrices b and c are symmetric, and that a + d^T = 0. Another equivalent condition is that A is of the form A = JS with S symmetric.Template:Rp

It follows easily from the definition that the transpose of a Hamiltonian matrix is Hamiltonian. Furthermore, the sum (and any linear combination) of two Hamiltonian matrices is again Hamiltonian, as is their commutator. It follows that the space of all Hamiltonian matrices is a Lie algebra, denoted sp(2n). The dimension of sp(2n) is 2n² + n. The corresponding Lie group is the symplectic group Sp(2n). This group consists of the symplectic matrices, those matrices Template:Mvar which satisfy A^TJA = J. Thus, the matrix exponential of a Hamiltonian matrix is symplectic. However the logarithm of a symplectic matrix is not necessarily Hamiltonian because the exponential map from the Lie algebra to the group is not surjective.Template:Rp

The characteristic polynomial of a real Hamiltonian matrix is even. Thus, if a Hamiltonian matrix has λ as an eigenvalue, then −λ, λ^* and −λ^* are also eigenvalues.Template:Rp It follows that the trace of a Hamiltonian matrix is zero.

The square of a Hamiltonian matrix is skew-Hamiltonian (a matrix Template:Mvar is skew-Hamiltonian if (JA)^T = −JA). Conversely, every skew-Hamiltonian matrix arises as the square of a Hamiltonian matrix.

Extension to complex matrices

As for symplectic matrices, the definition for Hamiltonian matrices can be extended to complex matrices in two ways. One possibility is to say that a matrix Template:Mvar is Hamiltonian if (JA)^T = JA, as above. Another possibility is to use the condition (JA)^* = JA where the superscript asterisk ((⋅)^*) denotes the conjugate transpose.

Hamiltonian operators

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

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

Choose a basis e₁, …, e_2n in Template:Mvar, 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.