Generalized Eigenvalues of $P_i^TLP_j+zP_i^TP_j$ - MathOverflow most recent 30 from http://mathoverflow.net2013-05-23T10:00:12Zhttp://mathoverflow.net/feeds/question/57219http://www.creativecommons.org/licenses/by-nc/2.5/rdfhttp://mathoverflow.net/questions/57219/generalized-eigenvalues-of-p-itlp-jzp-itp-jGeneralized Eigenvalues of $P_i^TLP_j+zP_i^TP_j$dan2011-03-03T08:07:14Z2011-03-03T08:07:14Z
<p>I have the following generalized eigenvalue problem</p>
<p>$\det[P_i^TLP_j+zP_i^TP_j] = 0$</p>
<p>$L$ 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.</p>
<p>What, if anything, can be said about the generalized eigenvalues $z$ of this problem?</p>