# Definition The Pfaffian of an antisymmetric (skew-symmetric) [[Square Matrix]] $A$ with [[dimension]] $2n \times 2n$ is a polynomial of degree $n$. It is given by: $ \text{pf}^2(A) = \det(A) $ where $\det(A)$ is the [[determinant]] of A. # Properties Below, $B$ is an arbitrary $2n \times 2n$ matrix $ \begin{align} \text{pf}(A^T) &= (-1)^n \text{pf}(A),\\ \text{pf}(\lambda A) &= (\lambda)^n \text{pf}(A),\\ \text{pf}^2(A) &= \det(A),\\ \text{pf}(BAB^T) &= \det(B) \text{pf}(A),\\ \text{pf}(A^{2m+1}) &= (-1)^{nm} \text{pf}(A),\\ \end{align} $ Assuming $A^T B$ is positive definite, we can write: $ \text{pf}(A) \text{pf}(B) = \exp\left(\frac{1}{2} \text{tr}(A^T B)\right)