Let $\bf x, y$ be two normalized eigenvectors of two different matrices. $\bf{x}$ is a Fiedler vector, so perpendicular to $\mathbf{1} := [1,1,\dots,1]$. What can you say about the following? $$k := \bf\frac{x^\top B \, y}{x^\top y}$$ Is $k\geq0$ for any condition or constraint?
My particular problem goes like this:
$A =B^{-1}C$
$A, C$ are singular matrices.
$Ay=\lambda y$,$Bz=\mu z$, $Cx=\sigma x$
$k=\frac{x^TBy}{x^Ty}$
I want some bounds on k here.