# 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

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### 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**

### 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: