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In [[mathematics]], the '''quadratic eigenvalue problem<ref>F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM
Rev., 43 (2001), pp. 235–286.</ref> (QEP)''', is to find [[scalar (mathematics)|scalar]] [[eigenvalue]]s <math>\lambda</math>, left [[eigenvector]]s <math>y</math> and right eigenvectors <math>x</math> such that
:<math> Q(\lambda)x = 0 ~ \text{ and } ~ y^\ast Q(\lambda) = 0,</math>
where <math>Q(\lambda)=\lambda^2
==Spectral theory==
A QEP is said to be <em>regular</em> if <math>\text{det} (Q(\lambda)) \not \equiv 0</math> identically. The coefficient of the <math>\lambda^{2n}</math> term in <math>\text{det}(Q(\lambda))</math> is <math>\text{det}(M)</math>, implying that the QEP is regular if <math>M</math> is nonsingular.
Eigenvalues at infinity and eigenvalues at 0 may be exchanged by considering the reversed polynomial, <math> \lambda^2 Q(\lambda^{-1}) = \lambda^2 K + \lambda C + M </math>. As there are <math> 2n</math> eigenvectors in a <math>n</math> dimensional space, the eigenvectors cannot be orthogonal. It is possible to have the same eigenvector attached to different eigenvalues.
==Applications==
=== Systems of differential equations ===
A QEP can result in part of the dynamic analysis of structures discretized by the [[finite element method]]. In this case the quadratic, <math>Q(\lambda)</math> has the form <math>Q(\lambda)=\lambda^2 M + \lambda C + K</math>, where <math>M</math> is the [[mass matrix]], <math>C</math> is the [[damping matrix]] and <math>K</math> is the [[stiffness matrix]].▼
Quadratic eigenvalue problems arise naturally in the solution of systems of second order [[Linear differential equation|linear differential equations]] without forcing:
Other applications include vibro-acoustics and fluid dynamics.▼
:<math> M q''(t) +C q'(t) + K q(t) = 0 </math>
Where <math> q(t) \in \mathbb{R}^n </math>, and <math> M, C, K \in \mathbb{R}^{n\times n}</math>. If all quadratic eigenvalues of <math> Q(\lambda) = \lambda^2 M + \lambda C + K </math> are distinct, then the solution can be written in terms of the quadratic eigenvalues and right quadratic eigenvectors as
:<math>
q(t) = \sum_{j=1}^{2n} \alpha_j x_j e^{\lambda_j t} = X e^{\Lambda t} \alpha
</math>
Where <math>\Lambda = \text{Diag}([\lambda_1, \ldots, \lambda_{2n}]) \in \mathbb{R}^{2n \times 2n} </math> are the quadratic eigenvalues, <math> X = [x_1, \ldots, x_{2n}] \in \mathbb{R}^{n \times 2n} </math> are the <math> 2n</math> right quadratic eigenvectors, and <math> \alpha = [\alpha_1, \cdots, \alpha_{2n}]^\top \in \mathbb{R}^{2n}</math> is a parameter vector determined from the initial conditions on <math> q</math> and <math> q'</math>.
[[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 [[Discretization|discretized]] by the [[finite element method]]. In this case the quadratic, <math>Q(\lambda)</math> has the form <math>Q(\lambda)=\lambda^2 M + \lambda C + K</math>, where <math>M</math> is the [[mass matrix]], <math>C</math> is the [[damping matrix]] and <math>K</math> is the [[stiffness matrix]].
▲Other applications include vibro-acoustics and [[fluid dynamics]].
==Methods of solution==
Direct methods for solving the standard or [[Generalized eigenvalue problem|generalized eigenvalue problems]] <math> Ax = \lambda x</math> and <math> Ax = \lambda B x </math>
are based on transforming the problem to [[Schur form|Schur]] or [[Schur decomposition#Generalized Schur decomposition|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]] (<math> A-\lambda B</math>), 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
:<math>
\lambda▼
\begin{bmatrix}
\end{bmatrix}
-
\lambda\begin{bmatrix}
\end{bmatrix},
</math>
:<math>
z =
\begin{bmatrix}
▲\lambda x
\end{bmatrix}.
</math>
For convenience, one often takes <math>N</math> to be the <math>n\times n</math> [[identity matrix]]. We solve <math> L(\lambda) z = 0 </math> for <math> \lambda </math> and <math>z</math>, for example by computing the Generalized Schur form. We can then
take the first <math>n</math> components of <math>z</math> as the eigenvector <math>x</math> of the original quadratic <math>Q(\lambda)</math>.
Another common linearization is given by
:<math>
L2(\lambda)= \begin{bmatrix}
-K & 0 \\
0 & N
\end{bmatrix}
-
\lambda\begin{bmatrix}
C & M \\
N & 0
\end{bmatrix}.
</math>
In the case when either <math>A</math> or <math>B</math> is a [[Hamiltonian matrix]] and the other is a [[skew-Hamiltonian matrix]], the following linearizations can be used.
:<math>
L3(\lambda)= \begin{bmatrix}
K & 0 \\
C & K
\end{bmatrix}
-
\lambda\begin{bmatrix}
0 & K \\
-M & 0
\end{bmatrix}.
</math>
:<math>
L4(\lambda)= \begin{bmatrix}
0 & -K \\
M & 0
\end{bmatrix}
-
\lambda\begin{bmatrix}
M & C \\
0 & M
\end{bmatrix}.
</math>
{{mathapplied-stub}}
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