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.

Examples

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 i_k < j_k. 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 = {a_ij} be a 2n×2n 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 associate to any antisymmetric 2n×2n matrix A ={a_ij} a bivector

where {e₁, e₂, …, e_2n} 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,

• Pf(A)² = det(A)

• Pf(BAB^T) = det(B)Pf(A)

• Pf(λA) = λⁿPf(A)

• Pf(A^T) = ( - 1)ⁿPf(A)

• For a block-diagonal matrix

we have Pf(A₁⊕A₂) = Pf(A₁)Pf(A₂).

• For an arbitrary n × n matrix 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.