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 polynomial matrix.
A QEP is said to be regularif identically. The coefficient of the term in is, implying that the QEP is regular if is nonsingular.
Eigenvalues at infinity and eigenvalues at 0 may be exchanged by considering the reversed polynomial, . As there are eigenvectors in a 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 , and . If all quadratic eigenvalues of are distinct, then the solution can be written in terms of the quadratic eigenvalues and right quadratic eigenvectors as
Where are the quadratic eigenvalues, are the right quadratic eigenvectors, and is a parameter vector determined from the initial conditions on and .
Stability theory for linear systems can now be applied, as the behavior of a solution depends explicitly on the (quadratic) eigenvalues.
Direct methods for solving the standard or generalized eigenvalue problems and
are based on transforming the problem to SchurorGeneralized 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 to be the identity matrix. 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 .