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 who indirectly named them after Johann Friedrich Pfaff. The Pfaffian (considered as a polynomial) is nonvanishing only for 2"n" × 2"n" 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 , a work based on earlier work on Pfaffian systems of ordinary differential equations by Jacobi.
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.
The Pfaffian of a 2"n" × 2"n" skew-symmetric tridiagonal matrix is given as
Formal definition.
Let "A" = ("a""i,j") be a 2"n" × 2"n" skew-symmetric matrix. The Pfaffian of "A" is explicitly defined by the formula
where "S"2"n" is the symmetric group of order (2"n")! and sgn(σ) is the signature of σ. Alternatively (but equivalently), the Pfaffian can be written also as
where the Levi-Civita symbol is used and summation over repeated indices is implied.
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, ..., 2"n"} into pairs without regard to order. There are (2"n")!/(2"n""n"!) = (2"n" - 1)!! such partitions. An element α ∈ Π can be written as
with "i""k" < "j""k" and formula_10. Let
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,
formula_14
and for "n" odd, this implies formula_15.
Recursive definition.
By convention, the Pfaffian of the 0×0 matrix is equal to one. The Pfaffian of a skew-symmetric 2"n"×2"n" matrix "A" with "n">0 can be computed recursively as
where index "i" can be selected arbitrarily, formula_17 is the Heaviside step function, and formula_18 denotes the matrix "A" with both the "i"-th and "j"-th rows and columns removed. Note how for the special choice formula_19 this reduces to the simpler expression:
Alternative definitions.
One can associate to any skew-symmetric 2"n"×2"n" matrix "A" =("a""ij") a bivector
where {"e"1, "e"2, ..., "e"2"n"} 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.
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 formula_23. For "m" odd, one can then show that this is equal to the usual Pfaffian of an ("m+"1) x ("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.
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.
Miscellaneous.
For a 2"n" × 2"n" skew-symmetric matrix "A"
For an arbitrary 2"n" × 2"n" matrix "B",
Substituting in this equation "B = Am", one gets for all integer "m"
Derivative identities.
If "A" depends on some variable "x""i", then the gradient of a Pfaffian is given by
and the Hessian of a Pfaffian is given by
Trace identities.
The product of the Pfaffians of skew-symmetric matrices "A" and "B" under the condition that "ATB" is a positive-definite matrix can be represented in the form of an exponential
Suppose "A" and "B" are "2n × 2n" skew-symmetric matrices, then
and "B"n("s"1,"s"2...,"s"n) are Bell polynomials.
Block matrices.
For a block-diagonal matrix
For an arbitrary "n" × "n" matrix "M":
It is often required to compute the pfaffian of a skew-symmetric matrix formula_36 with the block structure
where formula_38 and formula_39 are skew-symmetric matrices and formula_40 is a general rectangular matrix.
When formula_38 is invertible, one has
This can be seen from Aitken block-diagonalization formula,
This decomposition involves a congruence transformations that allow to use the pfaffian property formula_44.
Similarly, when formula_39 is invertible, one has
as can be seen by employing the decomposition 
Calculating the Pfaffian numerically.
Suppose "A" is a "2n × 2n" skew-symmetric matrices, then
where formula_49 is the second Pauli matrix, formula_50 is an identity matrix of dimension "n" and we took the trace over a matrix logarithm.
This equality is based on the trace identity
and on the observation that formula_52.
Since calculating the Logarithm of a matrix is a computationally demanding task, one can instead compute all eigenvalues of formula_53, take the log of all of these and sum them up. This procedure merely exploits the property formula_54. This can be implemented in Mathematica within a single line:
codice_1
For other efficient algorithms see .