Generalized Eigenvalues of $P_i^TLP_j+zP_i^TP_j$

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?

flag	asked Mar 3 2011 at 8:07 dan 237●10

Dear Dan: simultaneous cross posting between math.stackexchange and mathoverflow is discouraged, since there is quite an overlap of users. Please try to just pick the more appropriate forum and ask there. The Math.SE duplicate is: math.stackexchange.com/questions/24772/… – Willie Wong Mar 3 2011 at 9:19