---
tag: NH5HWBPWS9
kind: concept
name: Pfaffian
msc: [15A15]
sources:
  - url: https://en.wikipedia.org/wiki/Pfaffian
same_as:
  wikidata: Q1189744
  wikipedia: Pfaffian
---

## Summary

The Pfaffian is a polynomial in the entries of a [[skew-symmetric matrix|0PBY4E9GQF]] of even size whose square is the [[determinant|00KDZRW4C0]].
It is a sum over the [[perfect matchings|0NHK9MFFW9]] of the index set, with signs.
It appears in the study of alternating forms and symplectic groups, and in the counting of perfect matchings of planar graphs.

## Explanations

### L1 human

The determinant of a skew-symmetric matrix of even size is always a perfect square of a polynomial in its entries.
The Pfaffian is that square root, with a fixed choice of sign.
Each term of the Pfaffian pairs up the indices $1, \dots, 2n$ and multiplies the matching entries.
For $A = \begin{pmatrix} 0 & a \\ -a & 0 \end{pmatrix}$, the Pfaffian is $a$, and $\det A = a^2$.