Pfaffian

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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

[edit] Examples

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

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

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

[edit] 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 and $i_1 < i_2 < \cdots < i_n$ . 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.

[edit] 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.

[edit] 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:

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

[edit] 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.

The number of perfect matchings in a planar graph turns out to be the absolute value of a Pfaffian, hence is polynomial time computable. This is surprising given that for a general graph, the problem is very difficult (so called #P-complete). This result is used in physics to calculate the partition function of Ising models of spin glasses, where the underlying graph is planar. Recently it is also used to derive efficient algorithms for some otherwise seemingly intractable problems, including the efficient simulation of certain types of restricted quantum computation.

[edit] 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.