The identity $$\left| \begin{array}{cccc} x & y_1 & y_2 & y_3 \\ z_1 & 0 & a & b \\ z_2 & -a & 0 & c \\ z_3 & -b & -c & 0 \\ \end{array} \right|=\left(-a y_3+b y_2-c y_1\right) \left(a z_3-b z_2+c z_1\right)$$ is clearly a generalization of $$\left| \begin{array}{cccc} 0 & y_1 & y_2 & y_3 \\ -y_1 & 0 & a & b \\ -y_2 & -a & 0 & c \\ -y_3 & -b & -c & 0 \\ \end{array} \right|=\left(-a y_3+b y_2-c y_1\right)^2.$$ The same is true for similar $2n\times 2n$ matrices. Are these Pfaffian generalizations known?
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asked Dec 10, 2020 at 6:13
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4$\begingroup$ This is just the polarization identity $2Q(x,y) = Q(x+y,x+y) - Q(x,x) - Q(y,y)$ applied to both sides of the second equation, together with the fact that an odd-dimensional Pfaffian vanishes to get rid of the term on the diagonal. $\endgroup$– Bertram ArnoldCommented Dec 10, 2020 at 7:11
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