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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]] [[eigenvalue]]s λ<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(\lambda) = 0,\, </math>
where ''<math>Q''(λ\lambda) = λ<sup>\lambda^2</sup>''A''<sub>2</sub> A_2 + λ''A''<sub>1</sub> \lambda A_1 + ''A''<sub>0 A_0\,</submath>, with matrix coefficients ''A''<submath>2A_2\,</submath>, ''A''<submath>1A_1\,</submath> and ''A''<submath>0A_0\,</submath> that are of dimension ''<math>n''\,</math>-by-''<math>n''\,</math>. There are <math>2n\,</math> 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, ''<math>Q''(λ\lambda)\,</math> has the form λ<supmath>Q(\lambda)=\lambda^2</sup>'' M'' + λ'' \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.
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