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''of a skew-symmetric matrix $X$''
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$\def\Pf{\mathrm{Pf}\;}$''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
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The polynomial   $\Pf X$ in the entries of $X$ whose square is $\det X$.   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 $\Pf X$ 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},
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\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
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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
  
 
$$
 
$$
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A Pfaffian has the following properties:
 
A Pfaffian has the following properties:
  
#) $\text{Pf } (C^T X C) = (\det C) (\text{Pf } X)$ for any matrix $C$ of order $2n$;
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# $\Pf (C^T X C) = (\det C) (\Pf X)$ for any matrix $C$ of order $2n$;
 
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# $(\Pf X)^2= \det X$;
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072500/p07250023.png" />;
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# if $E$ is a [[Free module|free $A$-module]] with basis $e_1,\ldots,e_{2n}$ and if $$
 
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u = \sum_{i < j} x_{ij} e_i \wedge e_j \in \bigwedge^2 A,
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
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$$ then $$
 
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\bigwedge^n u =n! (\Pf X) e_1 \wedge \ldots \wedge e_{2n}.
<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>
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$$
 
 
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>
 
  
 
====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>
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<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>

Latest revision as of 10:09, 26 January 2012

$\def\Pf{\mathrm{Pf}\;}$of a skew-symmetric matrix $X$

The polynomial $\Pf X$ in the entries of $X$ whose square is $\det X$. 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 $\Pf X$ is the element of $A$ given by the formula

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

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

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:
Rafael.greenblatt/sandbox/Pfaffian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rafael.greenblatt/sandbox/Pfaffian&oldid=20488