Pfaffian

In mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries. This polynomial is called the Pfaffian of the matrix. The Pfaffian is nonvanishing only for 2n × 2n skew-symmetric matrices, in which case it is a polynomial of degree n.

Contents

Examples

<math>\mbox{Pf}\begin{bmatrix} 0 & a \\ -a & 0 \end{bmatrix}=a.<math>
<math>\mbox{Pf}\begin{bmatrix} 0 & a & b & c \\ -a & 0 & d & e \\ -b & -d & 0& f \\-c & -e & -f & 0 \end{bmatrix}=af-be+dc.<math>
<math>\mbox{Pf}\begin{bmatrix}

\begin{matrix}0 & \lambda_1\\ -\lambda_1 & 0\end{matrix} & 0 & \cdots & 0 \\ 0 & \begin{matrix}0 & \lambda_2\\ -\lambda_2 & 0\end{matrix} & & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \cdots & \begin{matrix}0 & \lambda_n\\ -\lambda_n & 0\end{matrix} \end{bmatrix} = \lambda_1\lambda_2\cdots\lambda_n.<math>

Formal definition

Let Π be the set of all partitions of {1, 2, …, 2n} into pairs without regard to order. There are (2n − 1)!! such partitions. An element α ∈ Π, can be written as

<math>\alpha=\{(i_1,j_1),(i_2,j_2),\cdots,(i_n,j_n)\}<math>

with ik < jk. Let

<math>\pi=\begin{bmatrix} 1 & 2 & 3 & 4 & \cdots & 2n \\ i_1 & j_1 & i_2 & j_2 & \cdots & j_{n} \end{bmatrix}<math>

be a corresponding permutation and let us define sgn(α) to be the signature of π. This depends only on the partition α and not on the particular choice of π.

Let A = {aij} be a 2n2n antisymmetric matrix. Given a partition α as above define

<math> A_\alpha =\operatorname{sgn}(\alpha)a_{i_1,j_1}a_{i_2,j_2}\cdots a_{i_n,j_n}.<math>

We can then define the Pfaffian of A to be

<math>\operatorname{Pf}(A)=\sum_{\alpha\in\Pi} A_\alpha.<math>

The Pfaffian of a n×n skew-symmetric matrix for n odd is defined to be zero.

Alternative definition

One can associate to any antisymmetric 2n2n matrix A ={aij} a bivector

<math>\omega=\sum_{i

where {e1, e2, …, e2n} is the standard basis of R2n. The Pfaffian is then defined by the equation

<math>\frac{1}{n!}\omega^n = \mbox{Pf}(A)\;e_1\wedge e_2\wedge\cdots\wedge e_{2n},<math>

here ωn denotes the wedge product of n copies of ω with itself.

Identities

For a 2n × 2n skew-symmetric matrix A and an arbitrary 2n × 2n matrix B,

  • <math>\mbox{Pf}(A)^2 = \det(A)<math>
  • <math>\mbox{Pf}(BAB^T)= \det(B)\mbox{Pf}(A)<math>
  • <math>\mbox{Pf}(\lambda A) = \lambda^n \mbox{Pf}(A)<math>
  • <math>\mbox{Pf}(A^T) = (-1)^n\mbox{Pf}(A)<math>
  • For a block-diagonal matrix
<math>A_1\oplus A_2=\begin{bmatrix} A_1 & 0 \\ 0 & A_2 \end{bmatrix}<math>
we have Pf(A1A2) = Pf(A1)Pf(A2).
  • For an arbitrary n × n matrix M:
<math>\mbox{Pf}\begin{bmatrix} 0 & M \\ -M^T & 0 \end{bmatrix} =

(-1)^{n(n-1)/2}\det M.<math>

Applications

The Pfaffian is an invariant polynomial of a skew-symmetric matrix (Note that it is not invariant under a general change of basis but rather under a proper orthogonal transformation). 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 which is used in the generalized Gauss-Bonnet theorem.

History

The term Pfaffian was introduced by Arthur Cayley, who used the term in 1852: "The permutants of this class (from their connection with the researches of Pfaff on differential equations) I shall term Pfaffians." The term honors German mathematician Johann Friedrich Pfaff.

ru:Пфаффиан

Navigation

  • Art and Cultures
    • Art (https://academickids.com/encyclopedia/index.php/Art)
    • Architecture (https://academickids.com/encyclopedia/index.php/Architecture)
    • Cultures (https://www.academickids.com/encyclopedia/index.php/Cultures)
    • Music (https://www.academickids.com/encyclopedia/index.php/Music)
    • Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
  • Biographies (http://www.academickids.com/encyclopedia/index.php/Biographies)
  • Clipart (http://www.academickids.com/encyclopedia/index.php/Clipart)
  • Geography (http://www.academickids.com/encyclopedia/index.php/Geography)
    • Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
    • Maps (http://www.academickids.com/encyclopedia/index.php/Maps)
    • Flags (http://www.academickids.com/encyclopedia/index.php/Flags)
    • Continents (http://www.academickids.com/encyclopedia/index.php/Continents)
  • History (http://www.academickids.com/encyclopedia/index.php/History)
    • Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
    • Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
    • Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
    • Prehistory (http://www.academickids.com/encyclopedia/index.php/Prehistory)
    • Renaissance (http://www.academickids.com/encyclopedia/index.php/Renaissance)
    • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
    • United States (http://www.academickids.com/encyclopedia/index.php/United_States)
    • Wars (http://www.academickids.com/encyclopedia/index.php/Wars)
    • World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
  • Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
  • Mathematics (http://www.academickids.com/encyclopedia/index.php/Mathematics)
  • Reference (http://www.academickids.com/encyclopedia/index.php/Reference)
  • Science (http://www.academickids.com/encyclopedia/index.php/Science)
    • Animals (http://www.academickids.com/encyclopedia/index.php/Animals)
    • Aviation (http://www.academickids.com/encyclopedia/index.php/Aviation)
    • Dinosaurs (http://www.academickids.com/encyclopedia/index.php/Dinosaurs)
    • Earth (http://www.academickids.com/encyclopedia/index.php/Earth)
    • Inventions (http://www.academickids.com/encyclopedia/index.php/Inventions)
    • Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
    • Plants (http://www.academickids.com/encyclopedia/index.php/Plants)
    • Scientists (http://www.academickids.com/encyclopedia/index.php/Scientists)
  • Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
    • Anthropology (http://www.academickids.com/encyclopedia/index.php/Anthropology)
    • Economics (http://www.academickids.com/encyclopedia/index.php/Economics)
    • Government (http://www.academickids.com/encyclopedia/index.php/Government)
    • Religion (http://www.academickids.com/encyclopedia/index.php/Religion)
    • Holidays (http://www.academickids.com/encyclopedia/index.php/Holidays)
  • Space and Astronomy
    • Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)
    • Planets (http://www.academickids.com/encyclopedia/index.php/Planets)
  • Sports (http://www.academickids.com/encyclopedia/index.php/Sports)
  • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
  • Weather (http://www.academickids.com/encyclopedia/index.php/Weather)
  • US States (http://www.academickids.com/encyclopedia/index.php/US_States)

Information

  • Home Page (http://academickids.com/encyclopedia/index.php)
  • Contact Us (http://www.academickids.com/encyclopedia/index.php/Contactus)

  • Clip Art (http://classroomclipart.com)
Toolbox
Personal tools