The Pfaffian is an analog of the determinant that is defined only for a 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.
Let be the set of all partition of into pairs of elements , can be represented as
with and , let
be a corresponding permutation and let us define to be the signature of a permutation ; clearly it depends only on the partition and not on the particular choice of . Given a partition as above let us set then we can define the Pfaffian of as
where denotes exterior product of copies of .
For any antisymmetric matrix ’ and any matrix
|Date of creation||2013-03-22 14:22:13|
|Last modified on||2013-03-22 14:22:13|
|Last modified by||PrimeFan (13766)|