I remember reading somewhere about the following properties of non-negative definite matrix. But I don't know how to prove it now.
Let $A$ and $B$ be two non-negative definite matrices. If $A^2\succ B^2$, then it necessarily follows that $A\succ B$, but $A\succ B$ doesn't necessarily leads to $A^2\succ B^2$.
How can you prove it? Thanks!