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# Pfaffian

The Pfaffian is an analog of the determinant that is defined only for a $2n\times 2n$ antisymmetric matrix. It is a polynomial of the polynomial ring in elements of the matrix, such that its square is equal to the determinant of the matrix.

Examples

$Pf\begin{bmatrix}0&a\\ -a&0\end{bmatrix}=a,$

$Pf\begin{bmatrix}0&a&b&c\\ -a&0&d&e\\ -b&-d&0&f\\ -c&-e&-f&0\end{bmatrix}=af-be+dc.$

Standard definition

Let

$A=\begin{bmatrix}0&a_{{1,2}}&\ldots&a_{{1,2n}}\\ -a_{{1,2}}&0&\ldots&a_{{2,2n}}\\ \vdots&\vdots&\vdots&\vdots\\ -a_{{2n,1}}&-a_{{2n,2}}&\ldots&0\end{bmatrix}.$ |

Let $\Pi$ be the set of all partition of $\{1,2,\ldots,2n\}$ into pairs of elements $\alpha\in\Pi$, can be represented as

$\alpha=\{(i_{1},j_{1}),(i_{2},j_{2}),\ldots,(i_{n},j_{n})\}$ |

with $i_{k}<j_{k}$ and $i_{1}<i_{2}<\cdots<i_{n}$, let

$\pi=\begin{bmatrix}1&2&3&4&\ldots&2n\\ i_{1}&j_{1}&i_{2}&j_{2}&\ldots&j_{{n}}\end{bmatrix}$ |

be a corresponding permutation and let us define
$sgn(\alpha)$ to be the signature of a permutation $\pi$; clearly it depends only on the partition $\alpha$ and not on the particular choice of $\pi$.
Given a partition $\alpha$ as above let us set
$a_{\alpha}=a_{{i_{1},j_{1}}}a_{{i_{2},j_{2}}}\ldots a_{{i_{n},j_{n}}},$
then we can define the *Pfaffian* of $A$ as

$Pf(A)=\sum_{{\alpha\in\Pi}}sgn(\alpha)a_{\alpha}.$ |

Alternative definition

One can associate to any antisymmetric $2n\times 2n$ matrix $A=\{a_{{ij}}\}$ a bivector :$\omega=\sum_{{i<j}}a_{{ij}}e_{i}\wedge e_{j}$ in a basis $\{e_{1},e_{2},\ldots,e_{{2n}}\}$ of $\mathbb{R}^{{2n}}$, then

$\omega^{n}=n!Pf(A)e_{1}\wedge e_{2}\wedge\cdots\wedge e_{{2n}},$ |

where $\omega^{n}$ denotes exterior product of $n$ copies of $\omega$.

Identities

For any antisymmetric $2n\times 2n$ matrix $A$’ and any $2n\times 2n$ matrix $B$

$Pf(A)^{2}=\det(A)$ |

$Pf(BAB^{T})=\det(B)Pf(A)$ |

## Mathematics Subject Classification

15A15*no label found*

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## Corrections

missing word by Mathprof ✓

missing space by Mathprof ✓

format by Mathprof ✓

missed one Pf by Mathprof ✓

## Comments

## Does HTML w/images show?

I don't know what I did wrong, but in HTML with images this entry shows just the word "Associates." Page images and TeX source look right. I'm using Mozilla 1.7.7 (Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.7))

## Re: Does HTML w/images show?

> I don't know what I did wrong, but in HTML with images this

> entry shows just the word "Associates." Page images and TeX

> source look right. I'm using Mozilla 1.7.7 (Mozilla/5.0

> (Windows; U; Windows NT 5.1; en-US; rv:1.7.7))

I get the same results with Firefox 1.5.0.5 (Mozilla/5.0 (X11; U; Linux i686; rv:1.8.0.5)).

Does rerendering the entry help?

## Re: Does HTML w/images show?

That did the trick. Thank you very much.