Quadratic Eigenvalue Problem

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In mathematics, the **quadratic eigenvalue problem (QEP)**, is to find scalar eigenvalues <math>lambda,</math>, left eigenvectors <math>y,</math> and right eigenvectors <math>x,</math> such that

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^</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.

## Methods of Solution

Direct methods for solving the standard or...

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- <math> Q(lambda)x = 0texty^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^</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

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