My books says that a unitary matrix $U$ can be written as:
$ U = A + i B$
where $A$ and $B$ are Hermitian and are related by:
$A^2 + B^2 = I$
where $I$ is the identity matrix. However, I cannot seem to find any proof of this statement and struggling to understand why that is the case.
Any square matrix $U$ with complex entries can be written in the form $U=A+iB$ with $A,B$ Hermitian by setting $$ A=\frac{U+U^*}{2}$$ and $$ B=\frac{U-U^*}{2i}$$ If also $U$ is unitary, then $U^*U=UU^*=I$, and it follows that $$ A^2+B^2=\frac{1}{4}(U+U^*)^2-\frac{1}{4}(U-U^*)^2=I$$
To understand the motivation behind the definitions of $A$ and $B$, it might be useful to think of the one dimensional case, when matrices are just complex numbers, Hermitian matrices correspond to real numbers, and unitary matrices to complex numbers with modulus one.