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
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)!

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: