= Skew-Hermitian decomposition of a unitary matrix
{title2=$U=A+iB,\quad A=(U-U^\dagger)/2,\quad B=(U+U^\dagger)/(2i)$}
The displayed parts of a <unitary matrix> are <skew-Hermitian matrices>. Expanding $UU^\dagger=U^\dagger U=I$ gives $AB=BA$ and $A^2+B^2=-I$. They admit a common <orthonormal eigenbasis>. For a unitary <eigenvalue> $u$, the corresponding part <eigenvalues> are $i\operatorname{Im}u$ and $-i\operatorname{Re}u$. If $U^T=-U$, then $U^\dagger=-\overline U$, so these are exactly its real and imaginary parts and both are real <skew-symmetric matrices>. Such an invertible antisymmetric matrix has even dimension, by its <determinant>. If one part has distinct <eigenvalues>, commutation alone shows its one-dimensional <eigenspaces> are preserved by the other part.
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