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
- <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 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.
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... Read More