Why is true that: If $A \succ 0$, $ A \succ B $ $\iff$ $ I -A^{-1/2}BA^{-1/2} \succ 0$ ??

Question

Why is this true??

If $A \succ 0$, $ A \succ B $ $\iff$ $ I -A^{-1/2}BA^{-1/2} \succ 0$

obs: I don't know for sure if one other hypothesis, $B \succeq 0$ ,is necessary here.

In the book, this claim is presented as follows:

Another question, I may present this in another topic if someone complains, where the hypothesis "$B \succeq 0$" was used in the proof above?

Answer 1

The author is using the fact that a positive definite matrix has a "square root" (i.e., $A^{1/2}$ is well-defined and invertible). Then, $$ A>B\iff A-B>0\iff\underbrace{A^{-1/2}AA^{-1/2}}_{I}-A^{-1/2}BA^{-1/2}>0. $$