The convention on skew-symmetric tridiagonal matrices, given below in the examples, then determines one specific polynomial, called the Pfaffian polynomial.
The term Pfaffian was introduced by Cayley (1852), who indirectly named them after Johann Friedrich Pfaff.
, which was first proved by Cayley (1849), 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 that 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.)
The Pfaffian of A is explicitly defined by the formula where S2n is the symmetric group of degree 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.
An element α ∈ Π can be written as with ik < jk and
Given a partition α as above, define The Pfaffian of A is then given by The Pfaffian of a n × n skew-symmetric matrix for n odd is defined to be zero, as the determinant of an odd skew-symmetric matrix is zero, since for a skew-symmetric matrix,
By convention, the Pfaffian of the 0 × 0 matrix is equal to one.
The Pfaffian of a skew-symmetric 2n × 2n matrix A with n > 0 can be computed recursively as where the index i can be selected arbitrarily,
denotes the matrix A with both the i-th and j-th rows and columns removed.
this reduces to the simpler expression: One can associate to any skew-symmetric 2n × 2n matrix A = (aij) a bivector where {e1, e2, ..., e2n} is the standard basis of R2n.
The Pfaffian is then defined by the equation here ω n denotes the wedge product of n copies of ω. Equivalently, we can consider the bivector (which is more convenient when we do not want to impose the summation constraint
A non-zero generalisation of the Pfaffian to odd-dimensional matrices is given in the work of de Bruijn on multiple integrals involving determinants.
[2] In particular for any m × m matrix A, we use the formal definition above but set
For m odd, one can then show that this is equal to the usual Pfaffian of an (m+1) × (m+1)-dimensional skew symmetric matrix where we have added an (m+1)th column consisting of m elements 1, an (m+1)th row consisting of m elements −1, and the corner element is zero.
The usual properties of Pfaffians, for example the relation to the determinant, then apply to this extended matrix.
is an equation of polynomials, it suffices to prove it for real matrices, and it would automatically apply for complex matrices as well.
By the spectral theory of skew-symmetric real matrices,
If A depends on some variable xi, then the gradient of a Pfaffian is given by and the Hessian of a Pfaffian is given by The product of the Pfaffians of skew-symmetric matrices A and B can be represented in the form of an exponential Suppose A and B are 2n × 2n skew-symmetric matrices, then and Bn(s1,s2,...,sn) are Bell polynomials.
is invertible, one has This can be seen from Aitken block-diagonalization formula,[3][4][5] This decomposition involves a congruence transformations that allow to use the Pfaffian property
is invertible, one has as can be seen by employing the decomposition Suppose A is a 2n × 2n skew-symmetric matrices, then where
This equality is based on the trace identity and on the observation that
This can be implemented in Mathematica with a single statement: However, this algorithm is unstable when the Pfaffian is large.
Under the summation, for a real valued Pfaffian, the argument of the exponential will be given in the form
is very large, rounding errors in computing the resulting sign from the complex phase can lead to a non-zero imaginary component.