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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
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