Square root of the determinant of a skew-symmetric square matrix
Not to be confused with Pfaffian function, Pfaffian system, or Pfaffian orientation.
In mathematics the determinant of an m×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. When m is even, it is a nonzero polynomial of degree m/2, and is unique up to multiplication by ±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 Cayley (1852), who indirectly named them after Johann Friedrich Pfaff.
Explicitly, for a skew-symmetric matrix ,
which was first proved by Cayley (1849), 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 minors and employing the recursion formula below.
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...
a constraint equation in Pfaffian form, whether the constraint is holonomic or nonholonomic depends on whether the Pfaffian form is integrable. See Universal...
In mathematics, Pfaffian functions are a certain class of functions whose derivative can be written in terms of the original function. They were originally...
In graph theory, a Pfaffian orientation of an undirected graph assigns a direction to each edge, so that certain cycles (the "even central cycles") have...
In dynamics, a Pfaffian constraint is a way to describe a dynamical system in the form: ∑ s = 1 n A r s d u s + A r d t = 0 ; r = 1 , … , L {\displaystyle...
A Pfaffian system is specified by 1-forms alone, but the theory includes other types of example of differential system. To elaborate, a Pfaffian system...
noted for his work on partial differential equations of the first order Pfaffian systems, as they are now called, which became part of the theory of differential...
convert the problem into a Pfaffian computation of a skew-symmetric matrix derived from a planar embedding of the graph. The Pfaffian of this matrix is then...
time and the coordinates but not on the momenta) Nonholonomic constraints Pfaffian constraint Scleronomic constraint (not depending on time) Rheonomic constraint...
by Catalan, Spottiswoode, Glaisher, and Scott; skew determinants and Pfaffians, in connection with the theory of orthogonal transformation, by Cayley;...
Investigations on the Foundations of Thermodynamics, which made use of Pfaffian systems and the concept of adiabatic accessibility, a notion that was introduced...
always +1 for any field. One way to see this is through the use of the Pfaffian and the identity Pf ( M T Ω M ) = det ( M ) Pf ( Ω ) . {\displaystyle...
Rote, G. (2001). "Division-free algorithms for the determinant and the pfaffian: algebraic and combinatorial approaches" (PDF). Computational discrete...
Photon Factory, a synchrotron located at KEK in Tsukuba, Japan pf(A), the Pfaffian of a matrix A Phenylphosphine, an organophosphorus compound Plasmodium...
Grassmann integral of a free Fermi field is a high-dimensional determinant or Pfaffian, which defines the new type of Gaussian integration appropriate for Fermi...
_{n}}} where Pf ( A ) {\displaystyle \operatorname {Pf} (A)} is the Pfaffian of A {\displaystyle A} and C = ( n 2 i ) {\textstyle {\mathcal {C}}={\binom...
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