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.

The Pfaffian is applied in the generalized Gauss-Bonnet theorem.

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)

Title	Pfaffian
Canonical name	Pfaffian
Date of creation	2013-03-22 14:22:13
Last modified on	2013-03-22 14:22:13
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	26
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 15A15