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PfaffianIn 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. Examples -
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\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 -
with ik < jk. Let -
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 2n2n antisymmetric matrix. Given a partition α as above define -
We can then define the Pfaffian of A to be -
The Pfaffian of a n×n skew-symmetric matrix for n odd is defined to be zero. Alternative definition One can asociate to any antisymmetric 2n2n matrix A ={aij} a bivector -
where {e1, e2, …, e2n} is the standard basis of R2n. The Pfaffian is then defined by the equation -
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, - For a block-diagonal matrix
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- we have Pf(A1⊕A2) = Pf(A1)Pf(A2).
- For an arbitrary n × n matrix M:
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(-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.
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