Quadratic eigenvalue problem

In mathematics, the quadratic eigenvalue problem (QEP), sometimes called a quadratic matrix equation, is to find scalar eigenvalues $\lambda \,$, left eigenvectors $y\,$ and right eigenvectors $x\,$ such that

$Q(\lambda )x=0{\text{ and }}y^{\ast }(\lambda )=0,\,$

where $Q(\lambda )=\lambda ^{2}A_{2}+\lambda A_{1}+A_{0}\,$, with matrix coefficients $A_{2}\,$, $A_{1}\,$ and $A_{0}\,$ that are of dimension $n\,$-by-$n\,$. There are $2n\,$ eigenvalues that may be infinite or finite, and possibly zero. This is a special case of a nonlinear eigenproblem.

Applications

A QEP can result in part of the dynamic analysis of structures discretized by the finite element method. In this case the quadratic, $Q(\lambda )\,$ has the form $Q(\lambda )=\lambda ^{2}M+\lambda C+K\,$, where $M\,$ is the mass matrix, $C\,$ is the damping matrix and $K\,$ is the stiffness matrix. Other applications include vibro-acoustics and fluid dynamics.