Quadratic eigenvalue problem: Difference between revisions

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In mathematics, the '''quadratic eigenvalue problem<ref>F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM

Rev., 43 (2001), pp. 235–286.</ref> (QEP)''', is to find [[scalar (mathematics)|scalar]] [[eigenvalue]]s <math>\lambda\,</math>, left [[eigenvector]]s <math>y\,</math> and right eigenvectors <math>x\,</math> such that

:<math> Q(\lambda)x = 0\text{ and }y^\ast Q(\lambda) = 0,\, </math>

where <math>Q(\lambda)=\lambda^2 A_2 + \lambda A_1 + A_0\,</math>, with matrix coefficients <math>A_2, \, A_1, A_0 \in \mathbb{C}^{n \times n}</math> and we require that <math>A_2\,\neq 0</math>, (so that we have a nonzero leading coefficient). There are <math>2n\,</math> eigenvalues that may be ''infinite'' or finite, and possibly zero. This is a special case of a [[nonlinear eigenproblem]]. <math>Q(\lambda)</math> is also known as a quadratic matrix polynomial.

==Applications==

A QEP can result in part of the dynamic analysis of structures discretized by the [[finite element method]]. In this case the quadratic, <math>Q(\lambda)\,</math> has the form <math>Q(\lambda)=\lambda^2 M + \lambda C + K\,</math>, where <math>M\,</math> is the [[mass matrix]], <math>C\,</math> is the [[damping matrix]] and <math>K\,</math> is the [[stiffness matrix]].

Other applications include vibro-acoustics and fluid dynamics.