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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 [[polynomial matrix]].
==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}(A_2)</math>, implying that the QEP is regular if <math>A_2</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 A_0 + \lambda A_1 + A_2 </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==
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