Define $S^n_{++}$ to be the set that contains all the positive definite matrices. That is, if $A \in S^n_{++}$, then $A$ is a positive definite matrix.
Then, we associate with each $A \in S^n_{++}$ an ellipsoid centered at the origin, given by
$$\varepsilon_A = \{x \mid x^T A^{-1} x \leq 1\}.$$
How to prove that for two positive definite matrices $A,B \in S^n_{++}$, $$B - A \in S^n_{++}$$ if and only if $$\varepsilon_A \subseteq \varepsilon_B?$$
This is related to Example 2.18 on Professor Boyd's textbook "Convex Optimization". I'm learning it by myself now.
We can use the answer here as follows: $$ B - A \in S^n_{++} \iff\\ I - B^{-1/2}AB^{-1/2} \in S^n_{++} \iff\\ I - A^{1/2}B^{-1}A^{1/2} \in S^n_{++} \iff\\ A^{-1} - B^{-1} \in S^n_{++} $$