# Generalization of Rayleigh quotient

Consider the function $$\sigma(x,y) = \frac{x^T A y}{x^T y}$$ where $$x, y \in \mathbb{R}^{d\times 1}$$ and $$A\in \mathbb{R}^{d\times d}$$. Also, its given that A is invertible with minimum eigenvalue $$\lambda$$.

Assume, $$x = [0 \ldots, 0, x_i, \ldots, x_d]$$ and $$y = [y_1, y_2, \ldots,x_i,\ldots, x_d]$$ for some fixed i. i.e. x contains some zero entries and rest non zero entries and is a subvector of y. What is $$\min_{x,y} \sigma(x,y)$$?