The Impact of the Properties of the Stiffness Matrix on Definite Quadratic Eigenvalue Problems - Kostić, Aleksandra; Voss, Heinrich; Timotić, Valentina (Sarajevo Journal of Mathematics, 2024)
Waiving the positive definiteness of the leading matrix $A$ in a hyperbolic quadratic eigenvalue problem $Q(λ)x=(λ^2A+λ B+C)x=0$, $x eq 0$ one obtains a definite eigenvalue problem, which is known to have $2n$ eigenvalues in $\mathbb R\cup\{∞\}$. One of the characterizations of the definite quadratic eigenvalue problem is the existence of parameters $ξ$ and $μ$ so that $Q(μ)$ is positive definite and $Q(ξ)$ is negative definite, where $ξ$ and $μ$ are not known in advance. In this paper we consider the impact of the properties of the stiffness matrix $C$ of the quadratic pencil $Q(λ)$ on the corresponding definite quadratic eigenvalue problem and on the localization of the parameters $ξ$ and $μ$.