Pfaffian

From Exampleproblems

In mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries. This polynomial is called the Pfaffian of the matrix. The Pfaffian is nonvanishing only for 2n × 2n skew-symmetric matrices, in which case it is a polynomial of degree n.

1 Examples
2 Formal definition
- 2.1 Alternative definition
3 Identities
4 Applications
5 History
6 External link

Examples

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

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

$\mbox{Pf}\begin{bmatrix} \begin{matrix}0 & \lambda_1\\ -\lambda_1 & 0\end{matrix} & 0 & \cdots & 0 \\ 0 & \begin{matrix}0 & \lambda_2\\ -\lambda_2 & 0\end{matrix} & & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \cdots & \begin{matrix}0 & \lambda_n\\ -\lambda_n & 0\end{matrix} \end{bmatrix} = \lambda_1\lambda_2\cdots\lambda_n.$

Formal definition

Let Π be the set of all partitions of {1, 2, …, 2n} into pairs without regard to order. There are (2n − 1)!! such partitions. An element α ∈ Π, can be written as

$\alpha=\{(i_1,j_1),(i_2,j_2),\cdots,(i_n,j_n)\}$

with i_k < j_k. Let

$\pi=\begin{bmatrix} 1 & 2 & 3 & 4 & \cdots & 2n \\ i_1 & j_1 & i_2 & j_2 & \cdots & j_{n} \end{bmatrix}$

be a corresponding permutation and let us define sgn(α) to be the signature of π. This depends only on the partition α and not on the particular choice of π.

Let A = {a_ij} be a 2n×2n skew-symmetric matrix. Given a partition α as above define

$A_\alpha =\operatorname{sgn}(\alpha)a_{i_1,j_1}a_{i_2,j_2}\cdots a_{i_n,j_n}.$

We can then define the Pfaffian of A to be

$\operatorname{Pf}(A)=\sum_{\alpha\in\Pi} A_\alpha.$

The Pfaffian of a n×n skew-symmetric matrix for n odd is defined to be zero.

Alternative definition

One can associate to any skew-symmetric 2n×2n matrix A ={a_ij} a bivector

$\omega=\sum_{i<j} a_{ij}\;e_i\wedge e_j.$

where {e₁, e₂, …, e_2n} is the standard basis of R²ⁿ. The Pfaffian is then defined by the equation

$\frac{1}{n!}\omega^n = \mbox{Pf}(A)\;e_1\wedge e_2\wedge\cdots\wedge e_{2n},$

here ωⁿ denotes the wedge product of n copies of ω with itself.

Identities

For a 2n × 2n skew-symmetric matrix A and an arbitrary 2n × 2n matrix B,

$Pf(A) 2 = det(A)$

$Pf(B A B T) = det(B)Pf(A)$

$Pf(λ A) = λ n Pf(A)$

$Pf(A T) = ( - 1) n Pf(A)$

For a block-diagonal matrix

$A_1\oplus A_2=\begin{bmatrix} A_1 & 0 \\ 0 & A_2 \end{bmatrix}$

we have Pf(A₁⊕A₂) = Pf(A₁)Pf(A₂).

For an arbitrary n × n matrix M:

$\mbox{Pf}\begin{bmatrix} 0 & M \\ -M^T & 0 \end{bmatrix} = (-1)^{n(n-1)/2}\det M.$

Applications

The Pfaffian is an invariant polynomial of a skew-symmetric matrix (Note that it is not invariant under a general change of basis but rather under a proper orthogonal transformation). As such, it is important in the theory of characteristic classes. In particular, it can be used to define the Euler class of a Riemannian manifold which is used in the generalized Gauss-Bonnet theorem.

History

The term Pfaffian was introduced by Arthur Cayley, who used the term in 1852: "The permutants of this class (from their connection with the researches of Pfaff on differential equations) I shall term Pfaffians." The term honors German mathematician Johann Friedrich Pfaff.