# Pfaffian

Pfaffianstrace identity
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.wikipedia
57 Related Articles

### Skew-symmetric matrix

skew-symmetricskew-symmetric matricesantisymmetric matrix
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.
This polynomial is called the Pfaffian of A and is denoted.

### Johann Friedrich Pfaff

Johann PfaffPfaffPfaff, Johann Friedrich
The term Pfaffian was introduced by who indirectly named them after Johann Friedrich Pfaff.
* Pfaffian

### Determinant

determinantsdetmatrix determinant
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.
Axisymmetric determinants have been studied by Lebesgue, Hesse, and Sylvester; persymmetric determinants by Sylvester and Hankel; circulants by Catalan, Spottiswoode, Glaisher, and Scott; skew determinants and Pfaffians, in connection with the theory of orthogonal transformation, by Cayley; continuants by Sylvester; Wronskians (so called by Muir) by Christoffel and Frobenius; compound determinants by Sylvester, Reiss, and Picquet; Jacobians and Hessians by Sylvester; and symmetric gauche determinants by Trudi.

### Chern–Gauss–Bonnet theorem

Chern theoremChern-Gauss-Bonnet theoremgeneralized Gauss–Bonnet theorem
where we have the Pfaffian.

### FKT algorithm

Fisher-Kasteleyn-Temperley algorithm
The key idea is to convert the problem into a Pfaffian computation of a skew-symmetric matrix derived from a planar embedding of the graph.

### Domino tiling

dimer modeldomino tilingsdimer
These numbers can be found by writing them as the Pfaffian of an skew-symmetric matrix whose eigenvalues can be found explicitly.

### Mathematics

mathematicalmathmathematician
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.

### Polynomial

polynomial functionpolynomialsmultivariate polynomial
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.

### Carl Gustav Jacob Jacobi

JacobiCarl Gustav JacobiCarl Jacobi
:which was first proved by, a work based on earlier work on Pfaffian systems of ordinary differential equations by Jacobi.

### Schur complement

then using induction and examining the Schur complement, which is skew symmetric as well.

### Tridiagonal matrix

tridiagonaltridiagonal matricestri-diagonal
The Pfaffian of a 2n × 2n skew-symmetric tridiagonal matrix is given as

### Symmetric group

symmetricinfinite symmetric groupS 4
where S 2n is the symmetric group of order (2n)!

### Parity of a permutation

even permutationodd permutationsignature
and sgn is the signature of σ.

### Permutation

permutationscycle notationpermuted
One can make use of the skew-symmetry of A to avoid summing over all possible permutations.

### Partition of a set

partitionpartitionspartitioned
Let Π be the set of all partitions of {1, 2, ..., 2n} into pairs without regard to order.

### Double factorial

generalizations of the double factorial!!214!!
There are (2n)!/(2 n n!) = (2n - 1)!! such partitions.

### Heaviside step function

Heaviside functionunit step functionHeaviside unit step function
where index i can be selected arbitrarily, \theta(i-j) is the Heaviside step function, and denotes the matrix A with both the i-th and j-th rows and columns removed.

### Exterior algebra

exterior productexterior powerwedge product
One can associate to any skew-symmetric 2n×2n matrix A ={a ij } a bivector

### Hessian matrix

HessianHessian determinantHessians
:and the Hessian of a Pfaffian is given by

### Definiteness of a matrix

positive definitepositive semidefinitepositive-definite
The product of the Pfaffians of skew-symmetric matrices A and B under the condition that A T B is a positive-definite matrix can be represented in the form of an exponential

### Bell polynomials

Bell polynomialcomplete Bell polynomialexponential Bell polynomials
:and B n (s 1,s 2,...,s n ) are Bell polynomials.

### Congruence relation

congruencecongruentcongruences
:This decomposition involves a congruence transformations that allow to use the pfaffian property.

### Pauli matrices

Pauli matrixPauli spin matricesPauli algebra
:where \sigma_y is the second Pauli matrix, I_n is an identity matrix of dimension n and we took the trace over a matrix logarithm.

### Logarithm of a matrix

matrix logarithmlogarithm
:where \sigma_y is the second Pauli matrix, I_n is an identity matrix of dimension n and we took the trace over a matrix logarithm.

### Wolfram Mathematica

MathematicaWolframwebMathematica
This can be implemented in Mathematica within a single line: