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| In mathematics, the '''quadratic eigenvalue problem<ref>F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM
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| 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
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| :<math> Q(\lambda)x = 0\text{ and }y^\ast Q(\lambda) = 0,\, </math>
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| where <math>Q(\lambda)=\lambda^2 A_2 + \lambda A_1 + A_0\,</math>, with matrix coefficients <math>A_2, \, A_1, A_0 \in \mathbb{C}^{n \times n}</math> and we require that <math>A_2\,\neq 0</math>, (so that we have a nonzero leading coefficient). There are <math>2n\,</math> eigenvalues that may be ''infinite'' or finite, and possibly zero. This is a special case of a [[nonlinear eigenproblem]]. <math>Q(\lambda)</math> is also known as a quadratic matrix polynomial.
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| ==Applications==
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| 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]].
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| Other applications include vibro-acoustics and fluid dynamics.
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| ==Methods of Solution==
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| Direct methods for solving the standard or generalized eigenvalue problems <math> Ax = \lambda x</math> and <math> Ax = \lambda B x </math>
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| are based on transforming the problem to [[Schur form|Schur]] or [[Generalized Schur form]]. However, there is no analogous form for quadratic matrix polynomials.
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| One approach is to transform the quadratic matrix polynomial to a linear [[matrix pencil]] (<math> A-\lambda B</math>), and solve a generalized
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| eigenvalue problem. Once eigenvalues and eigenvectors of the linear problem have been determined, eigenvectors and eigenvalues of the quadratic can be determined.
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| The most common linearization is the first companion linearization
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| :<math>
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| L(\lambda) =
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| \lambda
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| \begin{bmatrix}
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| M & 0 \\
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| 0 & I_n
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| \end{bmatrix}
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| +
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| \begin{bmatrix}
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| C & K \\
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| -I_n & 0
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| \end{bmatrix},
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| </math>
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| where <math>I_n</math> is the <math>n</math>-by-<math>n</math> identity matrix, with corresponding eigenvector | |
| :<math> | |
| z =
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| \begin{bmatrix}
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| \lambda x \\
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| x
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| \end{bmatrix}.
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| </math>
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| 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
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| 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>.
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| {{mathapplied-stub}}
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| ==References==
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| <references/>
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| [[Category:Linear algebra]]
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