Quadratic eigenvalue problem

In mathematics, the quadratic eigenvalue problem (QEP) is to find scalar eigenvalues λ, left eigenvectors y and right eigenvectors x such that

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

where Q(λ) = λ²A₂ + λA₁ + A₀, with matrix coefficients A₂, A₁ and A₀ that are of dimension n-by-n. There are 2n eigenvalues that may be infinite or finite, and possibly zero.

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(λ) has the form λ²M + λC + 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.