How to prove polar decomposition of $A$ using singular value decomposition?

Question

Consider Matrix $A_{n,n}$, show that there exists unitary matrix $Q$ and hermitian matrices $H1$ and $H2$ such that $A = H1Q = QH2$.

My start:

Using SVD,

$A = U \Sigma V^*$

$A = U I \Sigma V^*$

$A = U V^*V\Sigma V^*$

Since product of Unitary matrices is unitary, I can obtain $Q$, but how do I get $H1$ and $H2$?

Answer 1

Since $A=H_1Q$, you can right-multiply both sides by $Q^{-1}(=Q^\ast)$ to obtain $H_1$.