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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.

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  • $\begingroup$ By $C^\ast$ do you mean what is usually written $C^\dagger$, the hermitian adjoint of $C$, or $\bar C$, its complex conjugate? Note $C^\dagger = (\bar C)^T$. Regards. $\endgroup$ – Robert Lewis Dec 12 '14 at 18:35
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    $\begingroup$ @RobertLewis I do mean the complex conjugate. $\endgroup$ – Dia McThrees Dec 12 '14 at 18:36
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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 $x \in \Bbb C^n$.

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  • $\begingroup$ Why is $C^*(A-B)C$ hermitian? $\endgroup$ – Dia McThrees Dec 12 '14 at 18:40
  • $\begingroup$ @DiaMcThrees first of all: you don't need to accept the first answer given. In fact, if you need more guidance after getting that answer, I would recommend against doing so. $\endgroup$ – Omnomnomnom Dec 12 '14 at 18:41
  • $\begingroup$ @DiaMcThrees as for the question: show that $(C^*(A-B)C)^* = C^*(A-B)C$. Remember that $(PQ)^* = Q^*P^*$ $\endgroup$ – Omnomnomnom Dec 12 '14 at 18:42
  • $\begingroup$ Thank you. And $C^*(A-B)C$ is positive semidefinite because $A-B$ is, right? $\endgroup$ – Dia McThrees Dec 12 '14 at 18:44
  • $\begingroup$ Right. That is generally given as a theorem, but you should be comfortable with the proof of the fact that $M$ is PSD implies that $CMC^*$ is also PSD. $\endgroup$ – Omnomnomnom Dec 12 '14 at 18:46

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