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I have the following generalized eigenvalue problem

$\det[P_i^TLP_j+zP_i^TP_j]=0$

$L$ is a positive-semidefinite matrix with 1 eigenvalue at 0. More precisely, it is the combinatorial Laplacian matrix for a connected graph. $P_i$ ($P_j$) is the identity matrix with the $i$-th ($j$-th) column removed.

What, if anything, can be said about the generalized eigenvalues $z$ of this problem?

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    yes...$P_i$ is $n \times n-1$ matrix. $L$ has a real spectrum 0 = \lambda_1 < \lambda_2 \leq \lambda_3 \leq \ldots \leq \lambda_n; so yes, it has rank $n-1$, and the 0 eigenvalue has multiplicity one.2011-03-03

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