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.
Now suppose that $A,B \in S^n_{++}$ are two positive definite matrices. How to prove that $$B - A \in S^n_{++}$$ if and only if $$I - A^{1/2}B^{-1}A^{1/2} \in S^n_{++}?$$
The matrices $B-A$ and $I-B^{-1/2}AB^{-1/2}=I-X^*X$, where $X=A^{1/2}B^{-1/2}$, are congruent and thus one is positive definite if and only if the other is. Since $X^*X$ has the same eigenvalues as $XX^*=A^{1/2}B^{-1}A^{1/2}$, $B-A$ is positive definite if and only if $I-XX^*=I-A^{1/2}B^{-1}A^{1/2}$ is as well.