Pfaffian of a matrix

The Pfaffian is a polynomial associated with even-dimensional skew-symmetric matrices.
For a $2 m \times 2 m$ skew-symmetric matrix $A = (a_{i j})$ , the Pfaffian is denoted $Pf (A)$ and satisfies:

(Pf (A))^{2} = det (A)

Its explicit formula is:

Pf (A) := \frac{1}{2^{m} m!} \sum_{σ \in S_{2 m}} sgn (σ) \prod_{k = 1}^{m} a_{σ (2 k - 1), σ (2 k)} .

For odd-dimensional skew-symmetric matrices, the Pfaffian is defined to be zero.

A Pfaffian admits a Laplace-type expansion along any fixed row or column, closely analogous to the cofactor expansion of a determinant, but adapted to skew-symmetry and pairings.
Concretely, let $A = (a_{i j})$ be a $2 m \times 2 m$ skew‑symmetric matrix, and fix an index $j$ . Then the Pfaffian can be expanded along the $j$ ‑th column as

Pf (A) = \sum_{\begin{matrix} i = 1 \\ i \neq j \end{matrix}}^{2 m} (- 1)^{i + j + 1} a_{i j} Pf (A^{\hat{i}, \hat{j}}),

where $A^{\hat{i}, \hat{j}}$ denotes the $(2 m - 2) \times (2 m - 2)$ skew‑symmetric matrix obtained by deleting the $i$ ‑th and $j$ ‑th rows and columns of $A$ .

In Symplectic Geometry

In Symplectic Geometry, the Pfaffian gives a convenient way to express the contraction of vectors with the symplectic form. Let $ω$ be a symplectic form and $Ω = \frac{ω^{n}}{n!}$ the symplectic volume form. For vector fields $Y_{1}, \dots, Y_{2 n}$ , define the matrix $W$ with entries:

W_{a b} = ω (Y_{a}, Y_{b})

Then:

Ω (Y_{1}, \dots, Y_{2 n}) = Pf (W)