Pfaffian

The Pfaffian is a polynomial function associated with a skew-symmetric matrix that plays a central role in linear algebra and combinatorics. If A is skew-symmetric (A^T = -A) and of even size 2n, the Pfaffian pf(A) is defined so that pf(A)^2 = det(A). For odd-sized skew-symmetric matrices, the determinant is zero, and the Pfaffian is typically not defined in the same way (or taken as zero by convention since det(A) = 0).

For a 2n × 2n matrix, pf(A) can be written as a sum over all perfect matchings of the index set {1,...,2n}. Concretely, pf(A) = sum over partitions of {1,...,2n} into pairs {i1<j1},...,{in<jn} of sgn(π) ∏_{k=1}^n A_{ik, jk}, where sgn(π) is a suitable sign determined by the order of the pairs. A common, compact check is that for the 2 × 2 case with A = [ [0, a], [-a, 0] ], pf(A) = a.

Key properties include: pf(tA) = t^n pf(A) for a scalar t and 2n × 2n A; pf of a block diagonal matrix diag(B, C) equals pf(B) pf(C); and det(A) = pf(A)^2 for all skew-symmetric A of even size. The Pfaffian provides a computationally efficient way to handle determinants of skew-symmetric matrices and appears in several domains.

Applications span counting perfect matchings in graphs (via oriented adjacency matrices, such as Kasteleyn orientations), fermionic Gaussian integrals in physics, and various areas of geometry and representation theory where skew-symmetric forms arise.