A 
complex matrix 
is said to be Hamiltonian if
 |
(1)
|
where 
is the matrix of the form
![J_n=[0 I_n; I_n 0],](/images/equations/HamiltonianMatrix/NumberedEquation2.svg) |
(2)
|
is the 
identity matrix, and 
denotes the conjugate
transpose of a matrix 
. An analogous definition holds in the case of real 
matrices 
by requiring that 
be symmetric, i.e.,
by replacing 
by 
in (1).
Note that this criterion specifies precisely how a Hamiltonian matrix must look. Indeed, every Hamiltonian matrix 
(here assumed to have complex entries) must have the form
![A=[X D; G -X^H]](/images/equations/HamiltonianMatrix/NumberedEquation3.svg) |
(3)
|
where 
satisfy 
and 
.
This characterization holds for 
having strictly real entries as well by replacing all instances
of the conjugate transpose operator in (1)
by the transpose operator instead.
