In mathematics, the quadratic eigenvalue problem[1] (QEP), is to find scalar eigenvalues
, left eigenvectors
, (so that we have a nonzero leading coefficient).
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 polynomial matrix.
, implying that the QEP is regular if
Eigenvalues at infinity and eigenvalues at 0 may be exchanged by considering the reversed polynomial,
dimensional space, the eigenvectors cannot be orthogonal.
It is possible to have the same eigenvector attached to different eigenvalues.
Quadratic eigenvalue problems arise naturally in the solution of systems of second order linear differential equations without forcing: Where
are distinct, then the solution can be written in terms of the quadratic eigenvalues and right quadratic eigenvectors as Where
is a parameter vector determined from the initial conditions on
Stability theory for linear systems can now be applied, as the behavior of a solution depends explicitly on the (quadratic) eigenvalues.
A QEP can result in part of the dynamic analysis of structures discretized by the finite element method.
In this case the quadratic,
is the stiffness matrix.
Other applications include vibro-acoustics and fluid dynamics.
Direct methods for solving the standard or generalized eigenvalue problems
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 with corresponding eigenvector For convenience, one often takes
identity matrix.
, for example by computing the Generalized Schur form.
Another common linearization is given by In the case when either
is a Hamiltonian matrix and the other is a skew-Hamiltonian matrix, the following linearizations can be used.
This applied mathematics–related article is a stub.