`PfaffianDet`¶

The Pfaffian of an antisymmetric matrix, defined so that PfaffianDet[m]^2 == Det[m].

$ wo 'PfaffianDet[{{0, 1}, {-1, 0}}]'
1

$ wo 'PfaffianDet[{{0, 1, 2, 3}, {-1, 0, 4, 5}, {-2, -4, 0, 6}, {-3, -5, -6, 0}}]'
8

An odd-order matrix has Pfaffian 0:

$ wo 'PfaffianDet[{{0, 1, 2}, {-1, 0, 3}, {-2, -3, 0}}]'
0

A non-antisymmetric argument stays unevaluated:

$ wo 'PfaffianDet[{{5, 7}, {2, 9}}]'
PfaffianDet[{{5, 7}, {2, 9}}]

PfaffianDet¶