Quadratic eigenvalue problem

In mathematics, the quadratic eigenvalue problem (QEP), is to find scalar eigenvalues formula_1, left eigenvectors formula_2 and right eigenvectors formula_3 such that
where formula_5, with matrix coefficients formula_6 and we require that formula_7, (so that we have a nonzero leading coefficient). There are formula_8 eigenvalues that may be "infinite" or finite, and possibly zero. This is a special case of a nonlinear eigenproblem. formula_9 is also known as a quadratic polynomial matrix.
Applications.
A QEP can result in part of the dynamic analysis of structures discretized by the finite element method. In this case the quadratic, formula_9 has the form formula_11, where formula_12 is the mass matrix, formula_13 is the damping matrix and formula_14 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 formula_15 and formula_16
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 (formula_17), 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 formula_19 is the formula_20-by-formula_20 identity matrix, with corresponding eigenvector 
We solve formula_23 for formula_24 and formula_25, for example by computing the Generalized Schur form. We can then 
take the first formula_20 components of formula_25 as the eigenvector formula_3 of the original quadratic formula_9.