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Concept
Pfaffian
Summary
In mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer coefficients that only depend on the size of the matrix. The value of this polynomial, when applied to the coefficients of a skew-symmetric matrix, is called the Pfaffian of that matrix. The term Pfaffian was introduced by who indirectly named them after Johann Friedrich Pfaff. The Pfaffian (considered as a polynomial) is nonvanishing only for 2n × 2n skew-symmetric matrices, in which case it is a polynomial of degree n.
Explicitly, for a skew-symmetric matrix ,
which was first proved by , who cites Jacobi for introducing these polynomials in work on Pfaffian systems of differential equations. Cayley obtains this relation by specialising a more general result on matrices which deviate from skew symmetry only in the first row and the first column. The determinant of such a matrix is the product of the Pfaffians of the two matrices obtained by first setting in the original matrix the upper left entry to zero and then copying, respectively, the negative transpose of the first row to the first column and the negative transpose of the first column to the first row. This is proved by induction by expanding the determinant on minors and employing the recursion formula below.
(3 is odd, so the Pfaffian of B is 0)
The Pfaffian of a 2n × 2n skew-symmetric tridiagonal matrix is given as
(Note that any skew-symmetric matrix can be reduced to this form; see Spectral theory of a skew-symmetric matrix.)
Let A = (ai,j) be a 2n × 2n skew-symmetric matrix. The Pfaffian of A is explicitly defined by the formula
where S2n is the symmetric group of order (2n)! and sgn(σ) is the signature of σ.
One can make use of the skew-symmetry of A to avoid summing over all possible permutations. Let Π be the set of all partitions of {1, 2, ..., 2n} into pairs without regard to order. There are (2n)!/(2nn!) = (2n - 1)!! such partitions. An element α ∈ Π can be written as
with ik < jk and .
Official source
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