  A $2n\times 2n$ real matrix $A$ is said to be a Hamiltonian matrix if
$A^{T}J+JA=0$, where $J=\left( \begin{array}{cc} 0 & I_{n} \\ -I_{n} & 0\\
\end{array} \right)$. Hamiltonian matrices appear in many areas of
applications, such as linear control theory, linear equations in continuous
time systems, quadratic eigenvalue problems, and many other. In this paper we
study the inverse eigenvalue problerm for Hamiltonian matrices. In particular,
we give sufficient conditions for the existence and construction of a
Hamiltonian matrix with prescribed spectrum and we develop a Hamiltonian
version of a perturbation result, which allow us to change $r<2n$ eigenvalues
of a Hamiltonian matrix preserving its structure. Although our approach is of
theoretical nature, we also discuss an application of our results to the linear
continuous-time system through the bisection method.
