In mathematics, the quadratic eigenvalue problem (QEP), is to find scalar eigenvalues, left eigenvectors such that

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.

Spectral theory

A QEP is said to be regular is nonsingular.

dimensional space, the eigenvectors cannot be orthogonal. It is possible to have the same eigenvector attached to different eigenvalues.

Applications

Systems of differential equations

Quadratic eigenvalue problems arise naturally in the solution of systems of second order linear differential equations without forcing:

are distinct, then the solution can be written in terms of the quadratic eigenvalues and right quadratic eigenvectors as

. Stability theory for linear systems can now be applied, as the behavior of a solution depends explicitly on the (quadratic) eigenvalues.

Finite element methods

A QEP can result in part of the dynamic analysis of structures discretized by the finite element method is the mass matrix is the damping matrix 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 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

identity matrix.

Another common linearization is given by

is a Hamiltonian matrix and the other is a skew-Hamiltonian matrix, the following linearizations can be used.

References