\documentclass{article}
\usepackage{amsfonts}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/input pfaffian}
\author{Timothy Daly, Gunter Rote, Martin Rubey}
\maketitle
\begin{abstract}
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 $2n \times 2n$ skew-symmetric matrices, in which
case it is a polynomial of degree $n$.

\end{abstract}
\eject
\tableofcontents
\eject
\section{Examples}
$$
{\rm Pfaffian}\left[
\begin{array}{cc}
0 & a\\
-a & 0
\end{array}
\right] = a
$$

$$
{\rm Pfaffian}\left[
\begin{array}{cccc}
0 & a & b & c\\
-a & 0 & d & e\\
-b & -d & 0 & f\\
-c & -e & -f & 0
\end{array}
\right] = af -be + dc
$$

$$
{\rm Pfaffian}\left[
\begin{array}{cccc}
\begin{array}{cc}
0 & \lambda_1\\
-\lambda_1 & 0
\end{array} & 0 & \cdots & 0\\
0 &
\begin{array}{cc}
0 & \lambda_2\\
-\lambda_2 & 0
\end{array} & & 0\\
\vdots & & \ddots & \vdots\\
0 & 0 & \cdots &
\begin{array}{cc}
0 & \lambda_n\\
-\lambda_n & 0
\end{array}
\end{array}
\right] = \lambda_1\lambda_2\cdots\lambda_n
$$
\section{Formal definition}

Let $A = \{a_{i,j}\}$ be a $2n \times 2n$ skew-symmetric matrix.
The Pfaffian of A is defined by the equation

$$Pf(A) = \frac{1}{s^n n!}\sum_{\sigma\in{}S_{2n}}sgn(\sigma)
\prod_{i=1}^n{a_{\sigma(2i-1),\sigma(2i)}}$$

where $S_{2n}$ is the symmetric group and $sgn(\sigma)$
is the signature of $\sigma$.

One can make use of the skew-symmetry of $A$ to avoid summing over all
possible permutations. Let $\Pi$ be the set of all partitions of
$\{1, 2, \ldots, 2n\}$ into pairs without regard to order.
There are $(2n - 1)!!$ such partitions. An element
$\alpha \in \Pi$, can be written as

$$\alpha=\{(i_1,j_1),(i_2,j_2),\cdots,(i_n,j_n)\}$$

with $i_k < j_k$ and $i_1 < i_2 < \cdots < i_n$. Let

$$
\pi=
\left[
\begin{array}{cccccc}
1 & 2 & 3 & 4 & \cdots & 2n \\
i_1 & j_1 & i_2 & j_2 & \cdots & j_{n}
\end{array}
\right]
$$

be a corresponding permutation. This depends only on the partition $\alpha$
and not on the particular choice of $\Pi$. Given a partition $\alpha$ as above
define

$$A_\alpha =sgn(\pi)a_{i_1,j_1}a_{i_2,j_2}\cdots a_{i_n,j_n}$$

The Pfaffian of $A$ is then given by

$$Pf(A)=\sum_{\alpha\in\Pi} A_\alpha$$

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

\subsection{Alternative definition}

One can associate to any skew-symmetric $2n \times 2n$
matrix $A=\{a_{ij}\}$ a bivector

$$\omega=\sum_{i<j} a_{ij}~e^i\wedge e^j$$

where $\{e^1, e^2, \ldots,  e^{2n}\}$ is the standard basis of
$\mathbb{R}^{2n}$. The Pfaffian is then defined by the equation

$$\frac{1}{n!}\omega^n = \mbox{Pf}(A)~e^1\wedge e^2\wedge\cdots\wedge e^{2n}$$

here $\omega^n$ denotes the wedge product of $n$ copies of
$\omega^n$ with itself.

\subsection{Derivation from Determinant}

The Pfaffian can be derived from the determinant for a skew-symmetric
matrix $A$ as follows. Using Laplace's formula we can write the
determinant as

$$\det(A)=(-1)^{p+1}a_{p1}A_{p1} +
(-1)^{p+2}a_{p2}A_{p2}+\cdots+(-1)^{n+p}a_{pn}A_{pn}$$

where $A_{pi}$ is the $p,i$ minor matrix of the matrix $A$. We further
use Laplace's formula to note that

$$\det(A[A_{ij}]) = \vert A \vert^n$$

since this determinant is that of an $n \times n$ matrix whose only
non-zero elements are the diagonals (each with value det(A)) and
$[A_{ij}]$ is a matrix whose $i,j$th component is the corresponding $i,j$
minor matrix. In this way, following a proof by Parameswaran, we can
write the determinant as,

$$\det(A)\equiv\Delta_n=
\left|
\begin{array}{cccc}
a_{11}&a_{12}&\cdots&a_{1n}\\
a_{21}&a_{22}&\cdots&a_{2n}\\
\vdots&&&\vdots\\
a_{n1}&a_{n2}&\cdots&a_{nn}
\end{array}
\right|
$$

The minor of
$$
\left|
\begin{array}{cc}
a_{11}&a_{12}\\
a_{21}&a_{22}
\end{array}
\right|
$$
would be $\Delta_{n - 2}$. With this notation

$$\left|
\begin{array}{cccc}
1&0&\cdots&0\\
0&1&\cdots&0\\
a_{31}&a_{32}&\cdots&a_{3n}\\
\cdots&\cdots&\cdots&\cdots\\
a_{n1}&a_{n2}&\cdots&a_{nn}
\end{array}
\right|
\times
\left|
\begin{array}{cccc}
A_{11}&A_{21}&\cdots&A_{n1}\\
A_{12}&A_{22}&\cdots&A_{n2}\\
A_{13}&A_{23}&\cdots&A_{n3}\\
\cdots&\cdots&\cdots&\cdots\\
A_{1n}&A_{2n}&\cdots&A_{nn}
\end{array}
\right| =
\left|
\begin{array}{cc}
A_{11}&A_{21}\\
A_{12}&a_{22}
\end{array}
\right|
\Delta_n^{n-2}
$$

Thus

$$\Delta_{n-2}\Delta_n^{n-1}=
\left|
\begin{array}{cc}
A_{11}&A_{21}\\
A_{12}&a_{22}
\end{array}
\right|
\Delta_n^{n-2}
$$

Of course, it was only arbitrarily that we chose to remove the first
two rows, and more generically we can write

$$A_{rr}A_{ss} - A_{rs}A_{sr} = \Delta_{rs,rs}\Delta_n$$

where $\Delta_{rs,rs}$ is the determinant of the original matrix with
the rows $r$ and $s$, as well as the columns $r$ and $s$ removed. The
equation above simplifies in the skew-symmetric case to

$$A_{rs} = \sqrt{\Delta_{rs,rs}\Delta_n}$$

We now plug this back into the original formula for the determinant,

$$\Delta_n=
(-1)^{p+1}a_{p1}\sqrt{\Delta_{p1,p1}\Delta_n} +
(-1)^{p+2}a_{p2}\sqrt{\Delta_{p2,p2}\Delta_n} +
\cdots +
(-1)^{n+p}a_{pn}\sqrt{\Delta_{pn,pn}\Delta_n}
$$

or with slight manipulation,

$$\sqrt{\Delta_n}=(-1)^{p+1}
\left( \ a_{p1}\sqrt{\Delta_{p1,p1}} -
a_{p2}\sqrt{\Delta_{p2,p2}} +
\cdots +
(-1)^{n-1}a_{pn}\sqrt{\Delta_{pn,pn}} \
\right)
$$

The determinant is thus the square of the right hand side, and so we
identify the right hand side as the Pfaffian.

\section{Identities}

For a $2n \times 2n$ skew-symmetric matrix $A$ and an arbitrary
$2n \times 2n$ matrix B,
\begin{itemize}
\item $Pf(A)^2 = \det(A)$
\item $Pf(BAB^T) = \det(B)Pf(A)$
\item $Pf(\lambda{}A) = \lambda^nPf(A)$
\item $Pf(A^T) = ( - 1)^nPf(A)$
\item For a block-diagonal matrix
$$A_1\oplus A_2=
\left[
\begin{array}{cc}
A_1 & 0 \\
0 & A_2
\end{array}
\right]
$$
$$Pf(A1\oplus A2) = Pf(A_11)Pf(A_2)$$
\item For an arbitrary $n \times n$ matrix $M$:
$$\mbox{Pf}
\left[
\begin{array}{cc}
0 & M \\
-M^T & 0
\end{array}
\right] =
(-1)^{n(n-1)/2}\det M
$$
\end{itemize}
\section{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.

The number of perfect matchings in a planar graph turns out to be the
absolute value of a Pfaffian, hence is polynomial time
computable. This is surprising given that for a general graph, the
problem is very difficult (so called \#P-complete). This result is used
to calculate the partition function of Ising models of spin glasses in
physics, respectively of Markov random fields in machine learning
(Globerson and Jaakkola, 2007), where the underlying graph is
planar. Recently it is also used to derive efficient algorithms for
some otherwise seemingly intractable problems, including the efficient
simulation of certain types of restricted quantum computation.

The calculation of the number of possible ways to tile a standard
chessboard or 8-by-8 checkerboard with 32 dominoes is a simple example
of a problem which may be solved through the use of the Pfaffian
technique. There are 12,988,816 possible ways to tile a chessboard in
this manner. Specifically, 12988816 is the number of possible ways to
cover an 8-by-8 square with 32 1-by-2 rectangles. 12988816 is a square
number: $12988816 = 3604^2$). Note that 12988816 can be written in the
form: $2\times 1802^2 + 2\times 1802^2$,
where all the numbers have a digital root of 2.

More generally, the number of ways to cover a $2n \times 2n$ square with
$2n^2$ dominoes (as calculated independently by Temperley and M.E. Fisher and
Kasteleyn in 1961) is given by

$$
\prod_{j=1}^N
\prod_{k=1}^N
\left ( 4\cos^2 \frac{\pi j}{2n + 1} +
4\cos^2 \frac{\pi k}{2n + 1} \right )
$$

This technique can be applied in many mathematics-related subjects,
for example, in the classical, 2-dimensional computation of the
dimer-dimer correlator function in quantum mechanics.

\section{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.

\section{Axiom code}
I have hacked together an algorithm to compute a Pfaffian, using an algorithm
of Gunter Rote.  Currently it's only an .input script, but if it's useful for
somebody else than myself, we could make it a little more professional.

Martin

<<*>>=
)clear all

--S 1 of 9
B0 n == matrix [[(if i=j+1 and odd? j then -1 else _
                  if i=j-1 and odd? i then 1 else 0) _
                    for j in 1..n] for i in 1..n]
--R
--R                                                                   Type: Void
--E 1

--S 2 of 9
PfChar(lambda, A) ==
   n := nrows A
   (n = 2) => lambda^2 + A.(1,2)
   M := subMatrix(A, 3, n, 3, n)
   r := subMatrix(A, 1, 1, 3, n)
   s := subMatrix(A, 3, n, 2, 2)

   p := PfChar(lambda, M)
   d := degree(p, lambda)

   B := B0(n-2)
   C := r*B
   g := [(C*s).(1,1), A.(1,2), 1]
   if d >= 4 then
       B := M*B
       for i in 4..d by 2 repeat
           C := C*B
           g := cons((C*s).(1,1), g)
   g := reverse! g

   res := 0
   for i in 0..d by 2 for j in 2..d+2 repeat
       c := coefficient(p, lambda, i)
       for e in first(g, j) for k in 2..-d by -2 repeat
           res := res +  c * e * lambda^(k+i)

   res
--R
--R                                                                   Type: Void
--E 2

--S 3 of 9
pfaffian A == eval(PfChar(l, A), l=0)
--R
--R                                                                   Type: Void
--E 3

--S 4 of 9
m:Matrix(Integer):=[[0,15],[-15,0]]
--R
--R        + 0    15+
--R   (4)  |        |
--R        +- 15  0 +
--R                                                         Type: Matrix Integer
--E 4

--S 5 of 9
pfaffian m
--R
--R   (5)  15
--R                                                     Type: Polynomial Integer
--E 5

--S 6 of 9
(a,b,c,d,e,f):=(3,5,7,11,13,17)
--R
--R   (6)  17
--R                                                        Type: PositiveInteger
--E

--S 7 of 9
m1:Matrix(Integer):=[[0,a,b,c],[-a,0,d,e],[-b,-d,0,f],[-c,-e,-f,0]]
--R
--R        + 0    3     5    7 +
--R        |                   |
--R        |- 3   0     11   13|
--R   (7)  |                   |
--R        |- 5  - 11   0    17|
--R        |                   |
--R        +- 7  - 13  - 17  0 +
--R                                                         Type: Matrix Integer
--E 7

--S 8 of 9
m1ans:=a*f-b*e+d*c
--R
--R
--R   (8)  63
--R                                                        Type: PositiveInteger
--E 8

--S 9 of 9
pfaffian m1
--R
--R   Compiling function B0 with type PositiveInteger -> Matrix Integer
--R
--R   (9)  63
--R                                                     Type: Polynomial Integer
--E 9
)spool
)lisp (bye)
@
\eject
\begin{thebibliography}{99}
\bibitem{1} {\bf "http://en.wikipedia.org/wiki/Pfaffian"}
\bibitem{2} ``The statistics of dimers on a lattice, Part I'', Physica,
vol.27, 1961, pp.1209-25, P.W. Kasteleyn.
\bibitem{3} Propp, James (2004),
``Lambda-determinants and domino-tilings'', arXiv:math.CO/0406301.
\bibitem{4} Globerson, Amir and Tommi Jaakkola (2007),
``Approximate inference using planar graph decomposition'',
Advances in Neural Information Processing Systems 19, MIT Press.
\bibitem{5} ``The Games and Puzzles Journal'',
No.14, 1996, pp.204-5, Robin J. Chapman, University of Exeter
\bibitem{6} ``Domino Tilings and Products of Fibonacci and Pell numbers'',
Journal of Integer Sequences, Vol.5, 2002, Article 02.1.2,
James A. Sellers, The Pennsylvania State University
\bibitem{7} ``The Penguin Dictionary of Curious and Interesting Numbers'',
revised ed., 1997, ISBN 0-14-026149-4, David Wells, p.182.
\bibitem{8} ``A Treatise on the Theory of Determinants'',
1882, Macmillan and Co., Thomas Muir, Online
\bibitem{9} ``Skew-Symmetric Determinants'',
The American Mathematical Monthly, vol. 61, no.2., 1954, p.116,
S. Parameswaran Online-Subscription
\end{thebibliography}
\end{document}