Hermitian parts of a unitary matrix
= Hermitian parts of a unitary matrix
{c}
{title2=$A=(U+U^\dagger)/2,\quad B=(U-U^\dagger)/(2i)$}
Every <unitary matrix> has a unique decomposition $U=A+iB$ into <Hermitian matrices>. The formulas follow by adding and subtracting $U$ and its <conjugate transpose>. Expanding $U^\dagger U=UU^\dagger=I$ gives $A^2+B^2+i(AB-BA)=I$ and $A^2+B^2-i(AB-BA)=I$. Therefore the parts commute and $A^2+B^2=I$.