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Difference between revisions of "Pfaffian"

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''of a skew-symmetric matrix $X$''
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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 Pfaffian (of a [[skew-symmetric matrix]] $X$) is the  polynomial  $\def\Pf{\mathrm{Pf}\;} \Pf X$ in the entries of $X$ whose square is the [[determinant]] $\det X$.   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 \epsilon(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 $\epsilon(s)$ 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 $2n$;
# $(\text{Pf } X)^2= \det X$;
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# $(\Pf X)^2= \det X$;
# if $E$ is a free $A$-module with basis $e_1,\ldots,e_{2n}$ and if
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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,
$$
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$$ then $$
u = \sum_{i < j} x_{ij} e_i \bigwedge e_j \in \bigwedge^2 A,
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\bigwedge^n u =n! (\Pf X) e_1 \wedge \ldots \wedge e_{2n}.
$$
 
 
 
then
 
 
 
$$
 
\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>
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|align="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Elements of mathematics", '''2. Linear and multilinear algebra''', Addison-Wesley (1973) pp. Chapt. 2 (Translated from French)   {{MR|0274237}} 
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Latest revision as of 19:57, 30 November 2014

2020 Mathematics Subject Classification: Primary: 15-XX [MSN][ZBL]

The Pfaffian (of a skew-symmetric matrix $X$) is the polynomial $\def\Pf{\mathrm{Pf}\;} \Pf X$ in the entries of $X$ whose square is the determinant $\det X$. 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

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

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 $\epsilon(s)$ 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

[Bo] N. Bourbaki, "Elements of mathematics", 2. Linear and multilinear algebra, Addison-Wesley (1973) pp. Chapt. 2 (Translated from French) MR0274237
How to Cite This Entry:
Pfaffian. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pfaffian&oldid=20492
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article