= Unitary skew-diagonalization of an antisymmetric matrix
{title2=$U^TZU=\bigoplus_r z_r\begin{pmatrix}0&1\\-1&0\end{pmatrix}$}
A complex <matrix> $Z=-Z^T$ can be put by unitary congruence into two-by-two skew blocks and zero blocks; the magnitudes $|z_r|$ are its <singular values>, each repeated twice. One proof uses the <antilinear map> $A(v)=Z\bar v$. Because $Z^\dagger=-\bar Z$, $A^2=-ZZ^\dagger$. On a nonzero eigenspace of $ZZ^\dagger$ with <eigenvalue> $s^2$, any unit vector $v$ has orthogonal partner $A(v)/s$, and $A(A(v)/s)=-sv$. These orthonormal pairs give the blocks; the orthogonal complement is invariant, so the procedure repeats. Phases of the paired vectors set the block phases.
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