Semidefinite Hermitian Matrices If $A$ and $B$ are positive semidefinite matrices, we write $A\geq B$ in
case $A-B$ is positive semidefinite. Throughout this problem, $A$ and $B$ are
Hermitian matrices. Show that


*

*$C^{\ast}AC\geq C^{\ast}BC$ for all $C\in M_{n}(\mathbb{C)}$

*If $A\geq B$ and if $B$ is invertible, then so is $A$ and $B^{-1}\geq
A^{-1}$.


I don't know how to start with either of the problems. For the first one, I have $$
  C^*A^*C=AC^2,
$$ so $$
  AC^2\geq BC^2.
$$ I don't know if this is helpful.

Your approach for the first assumes that $A$ and $C$ commute.  So no, your approach here is not helpful in this case.
Hint: consider $C^*(A-B)C$.  Why is this matrix Hermitian?  Why is this matrix positive semidefinite?
Possible approach for the second problem: first, verify that $A$ must also be invertible.  Then, note that $A \geq B \iff x^*Ax \geq x^*Bx$ for all $x \in \Bbb C^n$.
