The Pfaffian is an analog of the determinant that is defined only for a $2n\times 2n$ antisymmetric matrix. It is a polynomial of the polynomial ring in elements of the matrix, such that its square is equal to the determinant of the matrix.

The Pfaffian is applied in the generalized Gauss-Bonnet theorem.

Examples

$Pf\begin{bmatrix} 0 & a \\ -a & 0 \end{bmatrix}=a,$

$Pf\begin{bmatrix} 0 & a & b & c \\ -a & 0 & d & e \\ -b & -d & 0& f \\-c & -e & -f & 0 \end{bmatrix}=af-be+dc.$

Standard definition

Let

$$A=\begin{bmatrix} 0 & a_{1,2} & \ldots & a_{1,2n} \\ -a_{1,2} & 0 & \ldots & a_{2,2n} \\ \vdots & \vdots & \vdots & \vdots \\-a_{2n,1} & -a_{2n,2} & \ldots & 0 \end{bmatrix}.$$

Let $\Pi^{}_{}$ be the set of all partition of $\{1,2, \ldots ,2n\}$ into pairs of elements $\alpha\in \Pi^{}_{}$ , can be represented as $$\alpha^{}_{}=\{(i_1,j_1),(i_2,j_2), \ldots ,(i_n,j_n)\} $$ with $i_k<j_k$ and $i_1 < i_2 < \cdots < i_n$ , let $$\pi=\begin{bmatrix} 1 & 2 & 3 & 4 & \ldots & 2n \\ i_1 & j_1 & i_2 & j_2 & \ldots & j_{n} \end{bmatrix}$$ be a corresponding permutation and let us define $sgn(\alpha)$ to be the signature of a permutation $\pi^{}_{}$ ; clearly it depends only on the partition $\alpha$ and not on the particular choice of $\pi^{}_{}$ . Given a partition $\alpha^{}_{}$ as above let us set $a_\alpha =a_{i_1,j_1}a_{i_2,j_2} \ldots a_{i_n,j_n},$ then we can define the Pfaffian of $A$ as $$Pf(A)=\sum_{\alpha\in \Pi} sgn(\alpha)a_\alpha.$$

Alternative definition

One can associate to any antisymmetric $2n\times 2n$ matrix $A=\{a_{ij}\}$ a bivector :$\omega=\sum_{i<j} a_{ij} e_i\wedge e_j$ in a basis $\{e_1,e_2, \ldots ,e_{2n}\}$ of $\mathbb{R}^{2n}$ , then $$\omega^n= n!Pf(A)e_1\wedge e_2\wedge \cdots \wedge e_{2n},$$ where $\omega^n_{}$ denotes exterior product of $n$ copies of $\omega^{}_{}$ .

Identities

For any antisymmetric $2n\times 2n$ matrix $A$ ' and any $2n\times 2n$ matrix $B$

$$Pf(A)^2 = \det(A)$$ $$Pf(BAB^T)= \det(B)Pf(A)$$