If $A$ and $B$ are positive definite, $A-B$ is positive definite, can we say $B^{-1}-A^{-1}$ is positive definite? I think this should be true but I don't know how to prove it.
It would be great if someone can offer help :) Thanks a lot!.
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If $A = I$ and $B = 2I$ then $B^{-1}-A^{-1} = -\dfrac{1}{2}I$, which is negative definite. Did you mean to include the condition that $A-B$ is positive definite? – JimmyK4542 Apr 16 '19 at 01:05
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You are right! Sorry I forgot to add it. I will edit the question now. – Yingying Li Apr 16 '19 at 01:06
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@JonasMeyer Thank you so much! Somehow I didn't find this question before! – Yingying Li Apr 16 '19 at 01:11