Hamiltonian Realization of spectra

Abstract

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.