Let $P(\lambda)=\lambda M−L\in \mathbb{R}^{n \times n}$ be a matrix pencil with symmetric nonsingular matrix $M$ and $L$ is a weighted Laplacian matrix of a connected graph. Clearly $(0,1_n)$ is an eigenpair of $P(\lambda)$ and suppose $(\lambda_i,x_i) \in \mathbb{R} \times \mathbb{R}^n,\,i=1:n−1$ be the eigenpairs of $P(\lambda)$ then show that $x_i$ can be chosen such that $x^T_iLx_j=0\, \forall \,1\leq i\neq j \leq n−1$ and $x^T_iLx_i=1$ where $1_n$ denotes the vector with all entries are 1.

2$\begingroup$ Xposted: math.stackexchange.com/q/3594612/339790 $\endgroup$ – Rodrigo de Azevedo Mar 25 at 15:40

$\begingroup$ @RodrigodeAzevedo I have asked this question in math stackexchange too. I am expecting some positive response on this query. $\endgroup$ – Saheb Mar 25 at 16:08

$\begingroup$ @Saheb Users are normally discouraged from opening two versions of the same question (even if they're on different sites in the SE network) $\endgroup$ – Omnomnomnom Mar 25 at 18:59

$\begingroup$ @Omnomnomnom I got your point. $\endgroup$ – Saheb Mar 29 at 6:55