What is "pfaffian"

n. (alternative case form of Pfaffian English)

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 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 A,

pf(A) = det(A), 
which was first proved by Thomas Muir in 1882 .

The fact that the determinant of any skew symmetric matrix is the square of a polynomial can be shown by writing the matrix as a block matrix, then using induction and examining the Schur complement, which is skew symmetric as well.