The Pfaffian has come up recently in a colleague's research. It happens equal to the square-root of the determinant (it is not defined this way, but in another way), but it is only defined on a 2n x 2n antisymmetric matrix. I began investigating it and I noticed this interesting property:
Of course, the determinant Det(A), which is the square of Pf(A), we know to be invariant under transpose, so that power of (-1)^n is quite fortunate! It will square away!
Also notice this: Taking the transpose of an antisymmetric matrix is equivalent to multiplying it by -1. That means that for some n Pf and -1 commute and for other n Pf and -1 anticommute! This may, therefore, be intimately tied to bosons and fermions!
