Pfaffian
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 Cayley (1852) 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,
which was first proved by Thomas Muir in 1882 (Muir 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.
(3 is odd, so 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 with all equal to zero, see Spectral theory of a skew-symmetric matrix..)
Let A = {ai,j} be a 2n × 2n skew-symmetric matrix. The Pfaffian of A is defined by the equation
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