View source for Pfaffian

''of a skew-symmetric matrix $X$''

The polynomial   $\text{Pf } X$ in the entries of $X$ whose square is $\text{det } X$.   More precisely, if $X = \|x_{ij}\|$ is a skew-symmetric matrix (i.e.   $x_{ij}=-x_{ji}$, $x_{ii}=0$; such a matrix is sometimes also called an   alternating matrix) of order $2n$ over a commutative-associative ring   $A$ with a unit, then $\text{Pf } X$ is the element of $A$ given by the   formula

$$
\text{Pf } X = \sum_s \varepsilon(s)x_{i_1j_1}\ldots x_{i_nj_n},
$$

where  the summation is over all possible partitions $s$ of the  set  $\{1,\ldots,2n\}$ into  non-intersecting pairs $\{i_\alpha,j_\alpha\}$,  where one may  suppose that $i_\alpha<j_\alpha$, $\alpha=1,\ldots,n$,  and where $\varepsilon(s)$ is the sign of  the permutation

$$
\left(
\begin{matrix}
1 & 2 & \ldots & 2n-1 & 2n \\
i_1 & j_1 & \ldots & i_n & j_n
\end{matrix}
\right).
$$

A Pfaffian has the following properties:

# $\text{Pf } (C^T X C) = (\det C) (\text{Pf } X)$ for any matrix $C$ of order $2n$;
# $(\text{Pf } X)^2= \det X$;
# if $E$ is a free $A$-module with basis $e_1,\ldots,e_{2n}$ and if $$
u = \sum_{i < j} x_{ij} e_i \bigwedge e_j \in \bigwedge^2 A,
$$ then $$
\bigwedge^n u =n! (\text{Pf } X) e_1 \bigwedge \ldots \bigwedge e_{2n}.
$$

====References====
<table><TR><TD   valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,     "Elements of mathematics. Algebra: Modules. Rings. Forms" , '''2''' ,   Addison-Wesley  (1975)  pp. Chapt.4;5;6  (Translated from   French)</TD></TR></table>