Unitary skew-diagonalization of an antisymmetric matrix ($U^{T}ZU={⨁}_r{z}_r\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$)

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 $Z^{†}=-\bar{Z}$, $A^2=-Z{Z}^{†}$. On a nonzero eigenspace of $Z{Z}^{†}$ 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.