Given an $n \times n$ matrix $A$.
(1) Show $AA^{*}$ is similar to $A^{*}A$ by singular value decomposition.
(2) Consider polar forms that $A=UP=QV^{*}$ in which $U$ is $m \times n$; $P$ is $n \times n$; $Q$ is $m \times m$ and $V$ is $n \times m$; also, $P$ and $Q$ are positive semidefinite.
Show that $P$ must be unitarily similar to Q.
(3) Show that if $PU=UP$ or $QV^{*}=V^{*}Q$, then $A$ must be normal.
My trial:
(1) By SVD, $A=U \Sigma V^{*}$ and $A^{*}=V\Sigma^{*} U^{*}$.
So, we have $AA^{*}=U \Sigma V^{*}V\Sigma^{*} U^{*}=U \Sigma \Sigma^{*} U^{*}$
$\Rightarrow U^{*}AA^{*}U=\Sigma \Sigma^{*}.$
Similarly, $A^{*}A=V\Sigma^{*} U^{*}U \Sigma V^{*}=V\Sigma^{*} \Sigma V^{*}$
$\Rightarrow V^{*}A^{*}AV=\Sigma^{*} \Sigma=\Sigma \Sigma^{*}.$
Lastly, we have $U^{*}AA^{*}U=V^{*}A^{*}AV \Rightarrow (VU^{*})AA^{*}(UV^{*})=A^{*}A.$
Since unitary matrices $UV^{*}$ is invertible, we conclude $A^{*}A$ is similar to $AA^{*}$.
May I know if I am correct for Question (1)
(2) Question 2. I do not have idea how to do. I just know eigenvalues of matrices $P$ and $Q$ are greater than or equal to $0$.
(3) Question 3. I do not have idea either. Should I Schur triangularization?
Thanks in advance. Would you please give few lines of answer on Qs2 and 3? as I am very new to this topic.
