I am working on a paper where some objects appears quiet frequently. I feel like they have a deeper meaning which can be explained but I don't have the intuition to understand them. I'm hoping that someone here can help.
Let $\eta \in \mathbb{R}^n$ be a unit vector and $H$ a positive definite matrix in $\mathbb{R}^{n\times n}$ (The Hessian matrix of a convex function, to be more precise).
Then, the objects I am trying to understand are: $\langle \eta, H \eta \rangle$ and $\langle \eta, H^{-1} \eta \rangle$. I would be glad about every bit of gained intuition. Thank you very much!