# Hamiltonian Realization of spectra

## TL;DR

This paper investigates the inverse eigenvalue problem for Hamiltonian matrices, providing conditions for constructing matrices with prescribed spectra and a perturbation method to modify eigenvalues while preserving structure, with applications to control systems.

## Contribution

It offers new theoretical results on constructing Hamiltonian matrices with specific spectra and a structured eigenvalue perturbation method, extending existing knowledge in the field.

## Key findings

- Provided sufficient conditions for Hamiltonian matrix construction with given spectra.
- Developed a Hamiltonian version of eigenvalue perturbation results.
- Discussed an application to linear continuous-time systems using the bisection method.

## 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.