Quadratic eigenvalue problem
Encyclopedia
In mathematics, the quadratic eigenvalue problem (QEP), is to find scalar
eigenvalues
, left eigenvectors 
and right eigenvectors 
such that
where
, with matrix coefficients 
and we require that 
, (so that we have a nonzero leading coefficient). There are 
eigenvalues that may be infinite or finite, and possibly zero. This is a special case of a nonlinear eigenproblem. 
is also known as a quadratic matrix polynomial.
. In this case the quadratic,
has the form 
, where 
is the mass matrix
,
is the damping matrix
and
is the stiffness matrix.
Other applications include vibro-acoustics and fluid dynamics.
and 
are based on transforming the problem to Schur or Generalized Schur form. However, there is no analogous form for quadratic matrix polynomials.
One approach is to transform the quadratic matrix polynomial to a linear matrix pencil (
), and solve a generalized
eigenvalue problem. Once eigenvalues and eigenvectors of the linear problem have been determined, eigenvectors and eigenvalues of the quadratic can be determined.
The most common linearization is the first companion linearization
where
is the 
-by-
identity matrix, with corresponding eigenvector
We solve
for 
and 
, for example by computing the Generalized Schur form. We can then
take the first
components of 
as the eigenvector 
of the original quadratic 
.
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
eigenvalues
, left eigenvectors and right eigenvectors such thatwhere
, with matrix coefficients and we require that , (so that we have a nonzero leading coefficient). There are eigenvalues that may be infinite or finite, and possibly zero. This is a special case of a nonlinear eigenproblem. 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 methodFinite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
. In this case the quadratic,
has the form , where is the mass matrixMass matrix
In computational mechanics, a mass matrix is a generalization of the concept of mass to generalized coordinates. For example, consider a two-body particle system in one dimension...
,
is the damping matrixDamping matrix
In applied mathematics, a damping matrix is a matrix corresponding to any of certain systems of linear ordinary differential equations.A damping matrix is defined as follows...
and
is the stiffness matrix.Other applications include vibro-acoustics and fluid dynamics.
Methods of Solution
Direct methods for solving the standard or generalized eigenvalue problems and are based on transforming the problem to Schur or Generalized Schur form. However, there is no analogous form for quadratic matrix polynomials.
One approach is to transform the quadratic matrix polynomial to a linear matrix pencil (
), and solve a generalizedeigenvalue problem. Once eigenvalues and eigenvectors of the linear problem have been determined, eigenvectors and eigenvalues of the quadratic can be determined.
The most common linearization is the first companion linearization
where
is the -by- identity matrix, with corresponding eigenvectorWe solve
for and , for example by computing the Generalized Schur form. We can thentake the first
components of as the eigenvector of the original quadratic .