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Wikipedia / Pfaffian

In mathematics, the determinant of an m-by-m skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer coefficients that only depends on m. When m is odd, the polynomial is zero, and when m is even, it is a nonzero polynomial of degree m/2, and is unique up to multiplication by &pm;1. The convention on skew-symmetric tridiagonal matrices, given below in the examples, then determines one specific polynomial, called the Pfaffian polynomial. The value of this polynomial, when applied to the entries 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.

which was first proved by , 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 minorss and employing the recursion formula below.

Examples

(3 is odd, so the Pfaffian of B is 0)

The Pfaffian of a 2n × 2n skew-symmetric tridiagonal matrix is equal to

(Note that any skew-symmetric matrix can be reduced to this form by conjugation with a unitary matrix; see Spectral theory of a skew-symmetric matrix.)

Formal definition

Let A = (aij) be a 2n × 2n 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 partitionss of into pairs without regard to order. There are such partitions. A permutation σ specifies a partition

Because of the skew-symmetry of A, all of the permutations specifying a given partition contribute the same amount to the sum. Thus the number of terms in the sum can be reduced by picking a single permutation for each possible partition. An element can be written as

with . 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 and for n.

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 can be computed recursively as

where the index i is the Heaviside step function denotes the matrix A with both the i-th and j-th rows and columns removed. this reduces to the simpler expression:

Alternative definitions

One can associate to any skew-symmetric 2n × 2n matrix a bivector

where is the standard basis of R2n. The Pfaffian is then defined by the equation

where ωn denotes the wedge product of n copies of ω.

Odd generalization

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 × m matrix A. 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.

Properties and identities

Pfaffians have the following properties, which are similar to those of determinants.

Multiplication of a row and a column by a constant is equivalent to multiplication of the Pfaffian by the same constant.
Simultaneous interchange of two different rows and corresponding columns changes the sign of the Pfaffian.
A multiple of a row and corresponding column added to another row and corresponding column does not change the value of the Pfaffian.

Using these properties, Pfaffians can be computed quickly, akin to the computation of determinants.

Miscellaneous

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

Derivative identities

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

Trace identities

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.

Block matrices

For a block-diagonal matrix

For an arbitrary n n matrix M:

with the block structure

is a general rectangular matrix.

is invertible, one has

This can be seen from Aitken block-diagonalization formula,

This decomposition involves a congruence transformations.

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

is the second Pauli matrix is an identity matrix of dimension n and we took the trace over a matrix logarithm.

This equality is based on the trace identity

Since calculating the logarithm of a matrix is a computationally demanding task, one can instead compute all eigenvalues, take the log of all of these and sum them up. This procedure merely exploits the property. This can be implemented in Mathematica with a single statement:

Pf[x_] := Module[{n = Dimensions[x] / 2}, I^(n^2) Exp[ 1/2 Total[ Log[Eigenvalues[ Dot[Transpose[KroneckerProduct[PauliMatrix[2], IdentityMatrix[n]]], x] ]]]]]

is very large, rounding errors in computing the resulting sign from the complex phase can lead to a non-zero imaginary component.

For other (more) efficient algorithms see .

Applications

The Pfaffian is an invariant polynomial of a skew-symmetric matrix under a proper orthogonal change of basis. As such, it is important in the theory of characteristic classes. In particular, it can be used to define the Euler class of a Riemannian manifold that is used in the generalized Gauss–Bonnet theorem.
There exist programs for the numerical computation of the Pfaffian on various platforms (Python, Matlab, Mathematica) .
The number of perfect matchings in a planar graph is given by a Pfaffian, hence is polynomial time computable via the FKT algorithm. This is surprising given that for general graphs, the problem is very difficult (so called #P-complete). This result is used to calculate the number of domino tilings of a rectangle, the partition function of Ising models in physics, or of Markov random fields in machine learning (; ), where the underlying graph is planar. It is also used to derive efficient algorithms for some otherwise seemingly intractable problems, including the efficient simulation of certain types of restricted quantum computation. See Holographic algorithm for more information.