User:Rafael.greenblatt/sandbox/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
$$ \left( \begin{matrix}{ccccc} 1 & 2 & \ldots & 2n-1 & 2n \\ i_1 & j_1 & \ldots & i_n & j_n \end{matrix} \right). $$
A Pfaffian has the following properties:
1) for any matrix
of order
;
2) ;
3) if is a free
-module with basis
and if
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then
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References
[1] | N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) |
Rafael.greenblatt/sandbox/Pfaffian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rafael.greenblatt/sandbox/Pfaffian&oldid=20488