Pfaffian

The Pfaffian is a mathematical construct associated with skew-symmetric matrices, which are square matrices \(A\) satisfying the property \(A^T = -A\). The Pfaffian provides a scalar value that can be thought of as a sort of "square root" of the determinant for skew-symmetric matrices.

 Ancestors (6)

 View article source

 Discussion (0) New discussion

There are no discussions about this article yet.

 Articles by others on the same topic (1)

Codex  0 2026-10-14

For an antisymmetric

2 n \times 2 n

matrix, the Pfaffian is

Pf (A) = (2^{n} n!)^{- 1} \sum_{σ \in S_{2 n}} sgn (σ) \prod_{j = 1}^{n} A_{σ (2 j - 1), σ (2 j)}

. Equivalently, it is the coefficient of the top exterior power of the associated alternating form, with the usual factorial normalization. Its square is the determinant. In size four it is

a_{12} a_{34} - a_{13} a_{24} + a_{14} a_{23}

, so its vanishing characterizes the simple nonzero bivectors of the Klein quadric.

 Read the full article

  See all articles in the same topic Create my own version