Pfaffian


The PfaffianMathworldPlanetmath is an analog of the determinantMathworldPlanetmath that is defined only for a 2n×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[0a-a0]=a,

Pf[0abc-a0de-b-d0f-c-e-f0]=af-be+dc.

Standard definition

Let

A=[0a1,2a1,2n-a1,20a2,2n-a2n,1-a2n,20].

Let Π be the set of all partitionMathworldPlanetmath of {1,2,,2n} into pairs of elements αΠ, can be represented as

α={(i1,j1),(i2,j2),,(in,jn)}

with ik<jk and i1<i2<<in, let

π=[12342ni1j1i2j2jn]

be a corresponding permutationMathworldPlanetmath and let us define sgn(α) to be the signaturePlanetmathPlanetmath of a permutation π; clearly it depends only on the partition α and not on the particular choice of π. Given a partition α as above let us set aα=ai1,j1ai2,j2ain,jn, then we can define the Pfaffian of A as

Pf(A)=αΠsgn(α)aα.

Alternative definition

One can associate to any antisymmetric 2n×2n matrix A={aij} a bivector :ω=i<jaijeiej in a basis {e1,e2,,e2n} of 2n, then

ωn=n!Pf(A)e1e2e2n,

where ωn denotes exterior product of n copies of ω.

For any antisymmetric 2n×2n matrix A’ and any 2n×2n matrix B

Pf(A)2=det(A)
Pf(BABT)=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