# Orthogonality condition of symmetric matrix pencil

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.

• – Rodrigo de Azevedo Mar 25 at 15:40
• @RodrigodeAzevedo I have asked this question in math stackexchange too. I am expecting some positive response on this query. – Saheb Mar 25 at 16:08
• @Saheb Users are normally discouraged from opening two versions of the same question (even if they're on different sites in the SE network) – Omnomnomnom Mar 25 at 18:59
• @Omnomnomnom I got your point. – Saheb Mar 29 at 6:55