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Difference between revisions of "User:Rafael.greenblatt/sandbox/Pfaffian"

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$$
 
$$
 
\left(
 
\left(
\begin{matrix}{ccccc}
+
\begin{matrix}
 
1 & 2 & \ldots & 2n-1 & 2n \\
 
1 & 2 & \ldots & 2n-1 & 2n \\
 
i_1 & j_1 & \ldots & i_n & j_n
 
i_1 & j_1 & \ldots & i_n & j_n
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A Pfaffian has the following properties:
 
A Pfaffian has the following properties:
  
1) <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250020.png" /> for any matrix <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250021.png" /> of order <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250022.png" />;
+
#) $\text{Pf } (C^T X C) = (\det C) (\text{Pf } X)$ for any matrix $C$ of order $2n$;
  
 
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250023.png" />;
 
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250023.png" />;

Revision as of 14:17, 25 January 2012

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:

  1. ) $\text{Pf } (C^T X C) = (\det C) (\text{Pf } X)$ for any matrix $C$ of order $2n$;

2) ;

3) if is a free -module with basis and if

then

References

[1] N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French)
How to Cite This Entry:
Rafael.greenblatt/sandbox/Pfaffian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rafael.greenblatt/sandbox/Pfaffian&oldid=20487