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''of a skew-symmetric matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p0725001.png" />''
+
''of a skew-symmetric matrix $X$''
  
The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p0725002.png" /> in the entries of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p0725003.png" /> whose square is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p0725004.png" />. More precisely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p0725005.png" /> is a skew-symmetric matrix (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p0725006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p0725007.png" />; such a matrix is sometimes also called an alternating matrix) of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p0725008.png" /> over a commutative-associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p0725009.png" /> with a unit, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250010.png" /> is the element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250011.png" /> given by the formula
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250012.png" /></td> </tr></table>
+
$$
 +
\text{Pf } X = \sum_s \varepsilon(s)x_{i_1j_1}\ldots x_{i_nj_n},
 +
$$
  
where the summation is over all possible partitions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250013.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250014.png" /> into non-intersecting pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250015.png" />, where one may suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250017.png" />, and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250018.png" /> is the sign of the permutation
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250019.png" /></td> </tr></table>
+
$$
 +
\left(
 +
\begin{matrix}
 +
1 & 2 & \ldots & 2n-1 & 2n \\
 +
i_1 & j_1 & \ldots & i_n & j_n
 +
\end{matrix}
 +
\right).
 +
$$
  
 
A Pfaffian has the following properties:
 
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" />;
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# $\text{Pf } (C^T X C) = (\det C) (\text{Pf } X)$ for any matrix $C$ of order $2n$;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250023.png" />;
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# $(\text{Pf } X)^2= \det X$;
  
3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250024.png" /> is a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250025.png" />-module with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250026.png" /> and if
+
3) if $E$ is a free $A$-module with basis $e_1,\ldots,e_{2n}$ and if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250027.png" /></td> </tr></table>
+
$$
 +
u = \sum_{i < j} x_{ij} e_i \bigwedge e_j \in \bigwedge^2 A,
 +
$$
  
 
then
 
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250028.png" /></td> </tr></table>
+
$$
 +
\bigwedge^n u =n! (\text{Pf } X) e_1 \bigwedge \ldots \bigwedge e_{2n}.
 +
$$
  
 
====References====
 
====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>
+
<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>

Revision as of 14:27, 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 $2n$;
  1. $(\text{Pf } X)^2= \det X$;

3) 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

[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:
Pfaffian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pfaffian&oldid=20491
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article