Quadratic eigenvalue problem

In mathematics, the quadratic eigenvalue problem^[1] (QEP) is to find scalar eigenvalues ${\displaystyle \lambda \,}$, left eigenvectors ${\displaystyle y\,}$ and right eigenvectors ${\displaystyle x\,}$ such that

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

where ${\displaystyle Q(\lambda )=\lambda ^{2}A_{2}+\lambda A_{1}+A_{0}\,}$, with matrix coefficients ${\displaystyle A_{2}\,}$, ${\displaystyle A_{1}\,}$ and ${\displaystyle A_{0}\,}$ that are of dimension ${\displaystyle n\,}$-by-${\displaystyle n\,}$. There are ${\displaystyle 2n\,}$ 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, ${\displaystyle Q(\lambda )\,}$  has the form ${\displaystyle Q(\lambda )=\lambda ^{2}M+\lambda C+K\,}$ , where ${\displaystyle M\,}$  is the mass matrix, ${\displaystyle C\,}$  is the damping matrix and ${\displaystyle K\,}$  is the stiffness matrix. Other applications include vibro-acoustics and fluid dynamics.

References

1. F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), pp. 235–286.