Hamiltonian matrix

Template:Short descriptionTemplate:Use dmy dates In mathematics, a Hamiltonian matrix is a Template:Math-by-Template:Math matrix Template:Mvar such that Template:Math 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 Template:Math is the Template:Mvar-by-Template:Mvar identity matrix. In other words, Template:Mvar is Hamiltonian if and only if Template:Math where Template:Math denotes the transpose. (Not to be confused with Hamiltonian (quantum mechanics))

Properties

Suppose that the Template:Math-by-Template:Math 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 Template:Math be Hamiltonian is equivalent to requiring that the matrices Template:Math and Template:Math are symmetric, and that Template:Math. Another equivalent condition is that Template:Math is of the form Template:Math with Template:Math 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 Template:Math. The dimension of Template:Math is Template:Math. The corresponding Lie group is the symplectic group Template:Math. This group consists of the symplectic matrices, those matrices Template:Mvar which satisfy Template:Math. 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 Template:Math as an eigenvalue, then Template:Math, Template:Math and Template:Math 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 Template:Math). 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 Template:Math, as above. Another possibility is to use the condition Template:Math where the superscript asterisk (Template:Math) denotes the conjugate transpose.

Hamiltonian operators

Let Template:Mvar be a vector space, equipped with a symplectic form Template:Math. A linear map $A:V\mapsto V$ is called a Hamiltonian operator with respect to Template:Math 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 Template:Math in Template:Mvar, such that Template:Math is written as $\sum _i{e}_i\wedge e_{n+i}$. A linear operator is Hamiltonian with respect to Template:Math if and only if its matrix in this basis is Hamiltonian.