The Impact of the Properties of the Stiffness Matrix on Definite Quadratic Eigenvalue Problems

Abstract

Waiving the positive definiteness of the leading matrix $\mathbf{A}$ in a hyperbolic quadratic eigenvalue problem $\mathbf{Q}(\lambda)\mathbf{x}=(\lambda^2\mathbf{A}+\lambda \mathbf{B}+\mathbf{C})\mathbf{x}=\mathbf{0}$, $\mathbf{x}\neq \mathbf{0}$ one obtains a definite eigenvalue problem, which is known to have $2n$ eigenvalues in $\mathbb R\cup\{\infty\}$.
One of the characterizations of the definite quadratic eigenvalue problem is the existence of parameters $\xi$ and $\mu$ so that $\mathbf{Q}(\mu)$ is positive definite and $\mathbf{Q}(\xi)$ is negative definite, where $\xi$ and $\mu$ are not known in advance. In this paper we consider the
impact of the properties of the stiffness matrix $\mathbf{C}$ of the quadratic pencil $\mathbf{Q}(\lambda)$ on the corresponding definite quadratic eigenvalue problem and on the localization of the parameters $\xi$ and $\mu$.