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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 $\min_{x,y} \sigma(x,y)$?

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