Q: Inequalities between different inner products defined by positive definite, symmetric matrices Let $A, B \in \mathbb{R}^{n \times n}$ be positive definite, symmetric matrices with eigenvalues larger or equal than $1$ and let $y \in \mathbb{R}^n$ be a normalized vector, i.e. $\lVert y \rVert = 1$. Furthermore, $y^T A B^{-1} y \geq y^Ty$ holds.
I was wondering if the following proposition is correct and if so, how to prove it:
$$y^T A B^{-1} y > y^T y$$ holds if and only if $$y^T A^{-1} y < y^T B^{-1} y$$ holds.
I am currently stuck and left without ideas. I tried to use the Cauchy-Schwarz inequality in different ways but failed. Any help is greatly appreciated!
