On 10/23/08, Fred lunnon <fred.lunnon@gmail.com> wrote:
The Parameswaran argument is summarised at http://en.wikipedia.org/wiki/Pfaffian WFL
I didn't find the argument presented in that Wikipedia page particularly perspicacious; the following seems more helpful, and incidentally clarifies the matter of the ambiguous sign. Given some skew-symmetric matrix [a_{ij}] of order n, reduce it thus: at stage k pivot about a pair of elements a_{2k,2k+1} = -a_{2k+1,2k}, reducing all remaining elements in both row and column 2k and 2k+1 to zero. Should the prospective pivot be zero, permute rows and columns equally to correct this, recording the sign of the permutation. If no nonzero pivot can be found, the Pfaffian equals zero: when n is odd, the final row and column being already zero as a result of the previous stage, the Pfaffian vanishes. Easily, each stage preserves the skew-symmetry, k = 0, 1, ... n/2-1. Finally, the matrix is reduced to an equivalent skew-symmetric bi-diagonal form, where only a_{2k,2k+1} = -a_{2k+1,2k} are nonzero. The determinant now has only a single nonzero term (\prod_k a_{2k,2k+1})^2; the sgined Pfaffian equals the accumulated permutation sign times \prod_k a_{2k,2k+1}. Fred Lunnon