How to prove positive semidefiniteness of two matrices through Schur Complement?

Question

Let matrices $X$ and $Y$ be positive semidefinite. Show that $X \succeq Y \succ 0$ is equivalent to $Y^{-1} \succeq X^{-1}$.

The teacher tells me that it is easy to prove via the Schur complement. But I could not get his idea. Could anyone help me and give me some intructions? Thanks in advance!

Answer 1

Hint. In my opinion, it is easier to prove that when $X,Y$ are positive definite, the following four statements are equivalent:

• $X\succeq Y$,
• $Y^{-1/2}XY^{-1/2}\succeq I$,
• $\left(Y^{-1/2}XY^{-1/2}\right)^{-1}\preceq I$,
• $X^{-1}\preceq Y^{-1}$.

Yet you can also follow your teacher's hint --- simply consider the Schur complements $M/Y^{-1}$ and $M/X$ for the matrix $$ M=\pmatrix{X&I\\ I&Y^{-1}}. $$