Solutions of a Quadratic Inverse Eigenvalue Problem for Damped Gyroscopic Second-Order Systems

Abstract

Given $k$ pairs of complex numbers and vectors (closed under conjugation), we consider the inverse quadratic eigenvalue problem of constructing $n\times n$ real matrices $M$, $D$, $G$, and $K$, where $M>0$, $K$ and $D$ are symmetric, and $G$ is skew-symmetric, so that the quadratic pencil $Q(\lambda )={\lambda }^{2}M+\lambda (D+G)+K$ has the given $k$ pairs as eigenpairs. First, we construct a general solution to this problem with $k\le n$. Then, with the special properties $D=0$ and , we construct a particular solution. Numerical results illustrate these solutions.