Inverse Eigenvalue Problem with Non-simple Eigenvalues for Damped Vibration Systems

Abstract

In this paper, we will present a general form of real and symmetric $n\times n$ matrices $M$, $C$ and $K$ for a quadratic inverse eigenvalue problem QIEP: $Q(\lambda) \equiv (\lambda^{2}M +\lambda C+K) x=0$, so that $Q(\lambda)$ has a prescribed set of $k$ eigenvalues with algebraic multiplicity $n_{i}$, $i=1,\cdots,k$ which $2n_{1}+2n_{2}+\cdots+2n_{l} +n_{l+1}+\cdots+n_{k}=2n)$. This paper generalizes the method of inverse problem for self-adjoint linear
pencils, to self-adjoint quadratic pencils $Q(\lambda)$. It is shown that this inverse problem involves certain free parameters. Via appropriate choice of free variables in the general form of QIEP, we solve a QIEP.