Pfafian 2004-05-14 00:46:54 2008-06-18 20:06:54 Definition antisymmetric matrix % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The {\em Pfaffian} is an analog of the determinant that is defined only for a $2n\times 2n$ antisymmetric matrix. It is a polynomial of the polynomial ring in elements of the matrix, such that its square is equal to the determinant of the matrix. The Pfaffian is applied in the generalized Gauss-Bonnet theorem. {\bf Examples} $Pf\begin{bmatrix} 0 & a \\ -a & 0 \end{bmatrix}=a,$ $Pf\begin{bmatrix} 0 & a & b & c \\ -a & 0 & d & e \\ -b & -d & 0& f \\-c & -e & -f & 0 \end{bmatrix}=af-be+dc.$ {\bf Standard definition} Let $$A=\begin{bmatrix} 0 & a_{1,2} & \ldots & a_{1,2n} \\ -a_{1,2} & 0 & \ldots & a_{2,2n} \\ \vdots & \vdots & \vdots & \vdots \\-a_{2n,1} & -a_{2n,2} & \ldots & 0 \end{bmatrix}.$$ Let $\Pi^{}_{}$ be the set of all partition of $\{1,2, \ldots ,2n\}$ into pairs of elements $\alpha\in \Pi^{}_{}$, can be represented as $$\alpha^{}_{}=\{(i_1,j_1),(i_2,j_2), \ldots ,(i_n,j_n)\} $$ with $i_k<j_k$ and $i_1 < i_2 < \cdots < i_n$, let $$\pi=\begin{bmatrix} 1 & 2 & 3 & 4 & \ldots & 2n \\ i_1 & j_1 & i_2 & j_2 & \ldots & j_{n} \end{bmatrix}$$ be a corresponding permutation and let us define $sgn(\alpha)$ to be the signature of a permutation $\pi^{}_{}$; clearly it depends only on the partition $\alpha$ and not on the particular choice of $\pi^{}_{}$. Given a partition $\alpha^{}_{}$ as above let us set $a_\alpha =a_{i_1,j_1}a_{i_2,j_2} \ldots a_{i_n,j_n},$ then we can define the \emph{Pfaffian} of $A$ as $$Pf(A)=\sum_{\alpha\in \Pi} sgn(\alpha)a_\alpha.$$ {\bf Alternative definition} One can associate to any antisymmetric $2n\times 2n$ matrix $A=\{a_{ij}\}$ a bivector :$\omega=\sum_{i<j} a_{ij} e_i\wedge e_j$ in a basis $\{e_1,e_2, \ldots ,e_{2n}\}$ of $\mathbb{R}^{2n}$, then $$\omega^n= n!Pf(A)e_1\wedge e_2\wedge \cdots \wedge e_{2n},$$ where $\omega^n_{}$ denotes exterior product of $n$ copies of $\omega^{}_{}$. {\bf Identities} For any antisymmetric $2n\times 2n$ matrix $A$' and any $2n\times 2n$ matrix $B$ $$Pf(A)^2 = \det(A)$$ $$Pf(BAB^T)= \det(B)Pf(A)$$