= Pfaffian
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{title2=$\operatorname{Pf}(A)^2=\det A$}
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For an antisymmetric $2n\times2n$ matrix, the Pfaffian is $\operatorname{Pf}(A)=(2^nn!)^{-1}\sum_{\sigma\in S_{2n}}\operatorname{sgn}(\sigma)\prod_{j=1}^n A_{\sigma(2j-1),\sigma(2j)}$. Equivalently, it is the coefficient of the top exterior power of the associated alternating form, with the usual factorial normalization. Its square is the <determinant>. In size four it is $a_{12}a_{34}-a_{13}a_{24}+a_{14}a_{23}$, so its vanishing characterizes the simple nonzero <bivectors> of the <Klein quadric>.
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