This is a tricky problem I encountered in my research. $A\in \mathbb{R}^{n\times n}, x,y\in \mathbb{R}^{n}$.

As known, since $A$ is a positive definite matrix, $A^{-1}$ is a positive definite matrix.

• If $x=y$, then $x^TA^{-1}y>0$ always holds.
• Since $A(i, j)>0$, then $x^TAy>0$ always holds. However, $A^{-1}(i, j)> 0$ doesn't hold and $x^TA^{-1}y>0$ as well.

Based on my experiments, I have reached the following views:

1. Since $x(i), y(i)>0$, $x^TA^{-1}y>0$ is likely to hold when $y(i)>x(i)$.
2. The diagonal elements of $𝐴$ are all positive, and their magnitudes are nearly $n$ times larger than those of the off-diagonal elements.

I want to figure out under what conditions on $A, x, y$ that $x^TA^{-1}y>0$ will always hold.

This is a tricky problem I encountered in my research. $A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$, i.e. $\forall 1\leq i \leq n, 1 \leq j\leq n, A(i, j)>0, x(i), y(i)>0$.

As known, since $A$ is a positive definite matrix, $A^{-1}$ is a positive definite matrix.

• If $x=y$, then $x^TA^{-1}y>0$ always holds.
• Since $A(i, j)>0$, then $x^TAy>0$ always holds. However, $A^{-1}(i, j)> 0$ doesn't hold and $x^TA^{-1}y>0$ as well.

Based on my experiments, I have reached the following views:

1. Since $x(i), y(i)>0$, $x^TA^{-1}y>0$ is likely to hold when $y(i)>x(i)$.
2. The diagonal elements of $𝐴$ are all positive, and their magnitudes are nearly $n$ times larger than those of the off-diagonal elements.

I want to figure out under what conditions on $A, x, y$ that $x^TA^{-1}y>0$ will always hold.