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Suppose we have a quadratic eigenvalue problem $(A_{0}+\lambda A_{1}+ \lambda^{2} A_{2})x=0$. I'd to know if there are conditions under which the problem is known to have a small number of distinct eigenvalues (say, 3 or 4). If it helps, I can assume that $A_{0}=I,A_{1}=I-J$ and that $A_{2}$ is negative semidefinite.

P.S. $J$ is the all-ones matrix.

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What is the matrix $J$? –  Federico Poloni Jan 15 '13 at 14:30
    
@FedericoPoloni: the all-ones matrix. –  Felix Goldberg Jan 15 '13 at 15:18

1 Answer 1

The following paper may help, Algorithms for the Nonlinear Eigenvalue Problem, SIAM Journal of Numer. Anal., vol. 10, No. 4, P.674.

Your equation may be cast into a linear eigenvalue problem, as $A_0$ is non-singular (see the matrix $B$ after eq.(1.5) in the same paper).

I think probably the problem becomes $\lambda I - B = 0$.

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