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

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?

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, $$A>B\iff A-B>0\iff\underbrace{A^{-1/2}AA^{-1/2}}_{I}-A^{-1/2}BA^{-1/2}>0.$$
• so, the author is multplying the inequality by $A^{-1/2}$ from both sizes? Ok, got this. And, the other part of the question, do you know where the hypothesis "B is +ve semidefinite" was used? Thank you very much. – gustavoreche Jul 3 '16 at 21:22