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.
Contents
- Examples
- Formal definition
- Recursive definition
- Alternative definitions
- Properties and identities
- Applications
- References
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.
Examples
(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
Formal definition
Let A = {ai,j} be a 2n × 2n skew-symmetric matrix. The Pfaffian of A is defined by the equation
where S2n is the symmetric group of the dimension (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
be the corresponding permutation. 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,
Recursive definition
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 index i can be selected arbitrarily,
Alternative definitions
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 ωn denotes the wedge product of n copies of ω.
A non-zero generalisation of the Pfaffian to odd dimensional matrices is given in the work of de Bruijn on multiple integrals involving determinants. In particular for any m x m matrix A, we use the formal definition above but set
Properties and identities
Pfaffians have the following properties, which are similar to those of determinants.
Using these properties, Pfaffians can be computed quickly, akin to the computation of determinants.
For a 2n × 2n skew-symmetric matrix A
For an arbitrary 2n × 2n matrix B,
Substituting in this equation B = Am, one gets for all integer m
For a block-diagonal matrix
For an arbitrary n × n matrix M:
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 two skew-symmetric matrices can be represented in the form of an exponential