1.I.2A

Let $U$ be an $n \times n$ unitary matrix $\left(U^{\dagger} U=U U^{\dagger}=I\right)$. Suppose that $A$ and $B$ are $n \times n$ Hermitian matrices such that $U=A+i B$.

Show that

(i) $A$ and $B$ commute,

(ii) $A^{2}+B^{2}=I$.

Find $A$ and $B$ in terms of $U$ and $U^{\dagger}$, and hence show that $A$ and $B$ are uniquely determined for a given $U$.

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