If $A, B$ be Hermitian and similar, $S = UQ$ then $A$ and $B$ are unitarily similar

Question

Let $A, B ∈ M_n$ be Hermitian and similar:$A=SBS^{ - 1} $.

If $S = UQ$ is a polar decomposition($U$ is unitary matrix and $Q$ is positive semidefinite matrix), why does $A$ and $B$ are unitarily similar via $U$.

Answer 1

Since $SBS^{-1}= UQBQ^{-1}U^{-1}$ and thus also $QBQ^{-1}$ is still self-adjoint, it follows that $Q^{-1}BQ=QBQ^{-1}$ or $BQ^2=Q^2 B$. This implies that $B,Q$ commute, so $QBQ^{-1}=B$.