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.
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Examples
- <math>\mbox{Pf}\begin{bmatrix} 0 & a \\ -a & 0 \end{bmatrix}=a.<math>
- <math>\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.<math>
- <math>\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.<math>
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
- <math>\alpha=\{(i_1,j_1),(i_2,j_2),\cdots,(i_n,j_n)\}<math>
with ik < jk. Let
- <math>\pi=\begin{bmatrix} 1 & 2 & 3 & 4 & \cdots & 2n \\ i_1 & j_1 & i_2 & j_2 & \cdots & j_{n} \end{bmatrix}<math>
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 = {aij} be a 2n×2n antisymmetric matrix. Given a partition α as above define
- <math> A_\alpha =\operatorname{sgn}(\alpha)a_{i_1,j_1}a_{i_2,j_2}\cdots a_{i_n,j_n}.<math>
We can then define the Pfaffian of A to be
- <math>\operatorname{Pf}(A)=\sum_{\alpha\in\Pi} A_\alpha.<math>
The Pfaffian of a n×n skew-symmetric matrix for n odd is defined to be zero.
Alternative definition
One can associate to any antisymmetric 2n×2n matrix A ={aij} a bivector
- <math>\omega=\sum_{i
where {e1, e2, …, e2n} is the standard basis of R2n. The Pfaffian is then defined by the equation
- <math>\frac{1}{n!}\omega^n = \mbox{Pf}(A)\;e_1\wedge e_2\wedge\cdots\wedge e_{2n},<math>
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,
- <math>\mbox{Pf}(A)^2 = \det(A)<math>
- <math>\mbox{Pf}(BAB^T)= \det(B)\mbox{Pf}(A)<math>
- <math>\mbox{Pf}(\lambda A) = \lambda^n \mbox{Pf}(A)<math>
- <math>\mbox{Pf}(A^T) = (-1)^n\mbox{Pf}(A)<math>
- For a block-diagonal matrix
- <math>A_1\oplus A_2=\begin{bmatrix} A_1 & 0 \\ 0 & A_2 \end{bmatrix}<math>
- we have Pf(A1⊕A2) = Pf(A1)Pf(A2).
- For an arbitrary n × n matrix M:
- <math>\mbox{Pf}\begin{bmatrix} 0 & M \\ -M^T & 0 \end{bmatrix} =
(-1)^{n(n-1)/2}\det M.<math>
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.