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This question was originally asked in (https://math.stackexchange.com/questions/4265063/computation-of-the-pfaffian-of-a-particular-matrix). I did not find any satisfactory answer there. So I am ...

Let $n$ be a positive integer, and let $M = (m_{ij})$ be a skew $2n \times 2n$ matrix. That is, we have $m_{ij} = -m_{ji}$ for $1 \leq i, j \leq 2n$. Then it is well-known that $$\det M = p(M)^2,$$ ...

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|=\...

Introduction: Let's assume we have a 2-form $\alpha=(1/2)\sum_{j,k=1}^n a_{jk}\ e_j\wedge e_k$, where $n=2m$, and $a_{jk}\in\mathbb C$. We know that $\alpha^{\wedge m}=\alpha\wedge\alpha\dots\wedge\...

Wikipedia presents a recursive definition of the Pfaffian of a skew-symmetric matrix as $$ \operatorname{pf}(A)=\sum_{{j=1}\atop{j\neq i}}^{2n}(-1)^{i+j+1+\theta(i-j)}a_{ij}\operatorname{pf}(A_{\hat{\...

Let $V$ be a Euclidean vector space and let $V^{\mathbb{C}} = V \oplus V$ be its complexification, with complex structure $$J = \begin{pmatrix} 0 & -\mathrm{id}\\ \mathrm{id} & 0 \end{pmatrix}....