Quadratic matrix polynomials with Hamiltonian spectrum and oscillatory damped systems

Abstract.

We consider quadratic matrix polynomials of the form $L(\lambda) = \lambda^{2}A + \epsilon\lambda B + C$, where $\epsilon$ is a real parameter, A is positive definite and B and C are symmetric. The main results of the paper are the characterization of the class of symmetric matrices B for which the spectrum of the polynomial is symmetric with respect to the imaginary axis and solutions of the corresponding differential equation oscillate in time. We also extend the results in [2] to allow us to study the asymptotic behaviour of the eigenvalues for large $\epsilon$.