Pfaffian Orientation Via FKT Algorithm Example

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. The value of this polynomial, when applied to the coefficients of a skew-symmetric matrix, is called the Pfaffian of that matrix. The term Pfaffian was introduced by who indirectly named them after Johann Friedrich Pfaff. The Pfaffian (considered as a polynomial) is nonvanishing only for 2''n'' × 2''n'' skew-symmetric matrices, in which case it is a polynomial of degree ''n''.

Explicitly, for a skew-symmetric matrix $A$,
:$\operatorname(A)^2=\det(A),$
which was first proved by , who cites Jacobi for introducing these polynomials in work on Pfaffian systems of differential equations. Caley obtains this relation by specialising a more general result on matrices which 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.

Examples

:$A=\begin 0 & a \\ -a & 0 \end.\qquad\operatorname(A)=a.$

:$B=\begin 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end.\qquad\operatorname(B)=0.$

(3 is odd, so the Pfaffian of B is 0)

:$\operatorname\begin 0 & a & b & c \\ -a & 0 & d & e \\ -b & -d & 0& f \\-c & -e & -f & 0 \end=af-be+dc.$

The Pfaffian of a 2''n'' × 2''n'' skew-symmetric tridiagonal matrix is given as
:$\operatorname\begin 0 & a_1 & 0 & 0\\ -a_1 & 0 & 0 & 0\\ 0 & 0 &0 & a_2 \\ 0 & 0 & -a_2 &0&\ddots\\ &&&\ddots&\ddots& \\ &&&& &0&a_n\\ &&&&&-a_n&0 \end = a_1 a_2\cdots a_n.$
(Note that any skew-symmetric matrix can be reduced to this form with all $b_i$ equal to zero; see Spectral theory of a skew-symmetric matrix.)

Formal definition

Let ''A'' = (''a''_''i,j'') be a 2''n'' × 2''n'' skew-symmetric matrix. The Pfaffian of ''A'' is explicitly defined by the formula

:$\operatorname(A) = \frac\sum_\operatorname(\sigma)\prod_^a_ \,,$

where ''S''_2''n'' is the symmetric group of order (2''n'')! and sgn(σ) is the signature of σ.

One can make use of the skew-symmetry of ''A'' to avoid summing over all possible permutations. Let Π be the set of all partitions of into pairs without regard to order. There are (2''n'')!/(2^''n''''n''!) = (2''n'' - 1) !! such partitions. An element α ∈ Π can be written as

:$\alpha=\$

with ''i''_''k'' < ''j''_''k'' and $i_1 < i_2 < \cdots < i_n$. Let

:$\pi_\alpha=\begin 1 & 2 & 3 & 4 & \cdots & 2n -1 & 2n \\ i_1 & j_1 & i_2 & j_2 & \cdots & i_n & j_ \end$

be the corresponding permutation. Given a partition α as above, define

:$A_\alpha =\operatorname(\pi_\alpha)a_a_\cdots a_.$

The Pfaffian of ''A'' is then given by

:$\operatorname(A)=\sum_ A_\alpha.$

The Pfaffian of a ''n''×''n'' skew-symmetric matrix for ''n'' odd is defined to be zero, as the determinant of an odd skew-symmetric matrix is zero, since for a skew-symmetric matrix,

$\det\,A = \det\,A^\text = \det\left(-A\right) = (-1)^n \det\,A,$

and for ''n'' odd, this implies $\det\,A = 0$.

Recursive definition

By convention, the Pfaffian of the 0×0 matrix is equal to one. The Pfaffian of a skew-symmetric 2''n''×2''n'' matrix ''A'' with ''n''>0 can be computed recursively as

:$\operatorname(A)=\sum_^(-1)^a_\operatorname(A_),$

where index ''i'' can be selected arbitrarily, $\theta(i-j)$ is the Heaviside step function, and $A_$ denotes the matrix ''A'' with both the ''i''-th and ''j''-th rows and columns removed. Note how for the special choice $i=1$ this reduces to the simpler expression:

:$\operatorname(A)=\sum_^(-1)^a_\operatorname(A_).$

Alternative definitions

One can associate to any skew-symmetric 2''n''×2''n'' matrix ''A'' =(''a''_''ij'') a bivector

:$\omega=\sum_ a_\;e_i\wedge e_j.$

where is the standard basis of R²ⁿ. The Pfaffian is then defined by the equation

:$\frac\omega^n = \operatorname(A)\;e_1\wedge e_2\wedge\cdots\wedge e_,$
here ω^''n'' denotes the wedge product of ''n'' copies of ω.

A non-zero generalisation of the Pfaffian to odd dimensional matrices is given in the work of de Bruijn on multiple integrals involving determinants. In particular for any ''m'' x ''m'' matrix ''A'', we use the formal definition above but set $n = \lfloor m/2\rfloor$. For ''m'' odd, one can then show that this is equal to the usual Pfaffian of an (''m+''1) x (''m+''1) dimensional skew symmetric matrix where we have added an (''m+''1)th column consisting of ''m'' elements 1, an (''m+''1)th row consisting of ''m'' elements -1, and the corner element is zero. The usual properties of Pfaffians, for example the relation to the determinant, then apply to this extended matrix.

Properties and identities

Pfaffians have the following properties, which are similar to those of determinants.
* Multiplication of a row and a column by a constant is equivalent to multiplication of the Pfaffian by the same constant.
* Simultaneous interchange of two different rows and corresponding columns changes the sign of the Pfaffian.
* A multiple of a row and corresponding column added to another row and corresponding column does not change the value of the Pfaffian.
Using these properties, Pfaffians can be computed quickly, akin to the computation of determinants.

Miscellaneous

For a 2''n'' × 2''n'' skew-symmetric matrix ''A''
:$\operatorname(A^\text) = (-1)^n\operatorname(A).$
:$\operatorname(\lambda A) = \lambda^n \operatorname(A).$
:$\operatorname(A)^2 = \det(A).$
For an arbitrary 2''n'' × 2''n'' matrix ''B'',
:$\operatorname(BAB^\text)= \det(B)\operatorname(A).$
Substituting in this equation ''B = A^m'', one gets for all integer ''m''
:$\operatorname(A^)= (-1)^\operatorname(A)^.$

Derivative identities

If ''A'' depends on some variable ''x''_''i'', then the gradient of a Pfaffian is given by
:$\frac\frac=\frac\operatorname\left(A^\frac\right),$
and the Hessian of a Pfaffian is given by
:$\frac\frac=\frac\operatorname\left(A^\frac\right)-\frac\operatorname\left(A^\fracA^\frac\right)+\frac\operatorname\left(A^\frac\right)\operatorname\left(A^\frac\right).$

Trace identities

The product of the Pfaffians of skew-symmetric matrices ''A'' and ''B'' can be represented in the form of an exponential
:$\textrm(A)\,\textrm(B) = \exp(\tfrac\mathrm\log(A^\textB)).$

Suppose ''A'' and ''B'' are ''2n × 2n'' skew-symmetric matrices, then
:$\mathrm(A)\,\mathrm(B) = \tfrac B_n(s_1, s_2, \ldots, s_n), \qquad \mathrm \qquad s_l = - \tfrac(l - 1)!\,\mathrm((AB)^l)$
and ''B''_n(''s''₁,''s''₂,...,''s''_n) are Bell polynomials.

Block matrices

For a block-diagonal matrix
::$A_1\oplus A_2=\begin A_1 & 0 \\ 0 & A_2 \end,$
:$\operatorname(A_1\oplus A_2) =\operatorname(A_1)\operatorname(A_2).$
For an arbitrary ''n'' × ''n'' matrix ''M'':
:$\operatorname\begin 0 & M \\ -M^\text & 0 \end = (-1)^\det M.$

It is often required to compute the pfaffian of a skew-symmetric matrix $S$ with the block structure
:$S = \begin M & Q\\-Q^T & N \end\,$
where $M$ and $N$ are skew-symmetric matrices and $Q$ is a general rectangular matrix.

When $M$ is invertible, one has
:$\operatorname( S) = \operatorname( M)\operatorname( N+ Q^T M^ Q).$
This can be seen from Aitken block-diagonalization formula,Bunch, James R. "A note on the stable decomposition of skew-symmetric matrices." Mathematics of Computation 38.158 (1982): 475-479.
:$\beginM& 0\\ 0 & N+Q^T M^ Q\end = \beginI& 0\\ Q^T M^ & I\end\begin M& Q\\ -Q^T & N\end \beginI& -M^ Q\\ 0& I \end.$
This decomposition involves a congruence transformations that allow to use the pfaffian property $\operatorname(BAB^T) = \operatorname(B)\operatorname(A)$.

Similarly, when $N$ is invertible, one has
:$\operatorname( S) = \operatorname(N)\operatorname( M+ Q N^ Q^T),$
as can be seen by employing the decomposition
:$\beginM+Q N^ Q^T& 0\\ 0 & N\end = \beginI& -Q N^\\ 0 & I\end\begin M& Q\\ -Q^T & N\end \beginI& 0\\ N^ Q^T& I \end.$

Calculating the Pfaffian numerically

Suppose ''A'' is a ''2n × 2n'' skew-symmetric matrices, then
:$\textrm(A) = i^ \exp\left(\tfrac\mathrm\log((\sigma_y\otimes I_n)^T\cdot A)\right),$
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.

This equality is based on the trace identity
:$\textrm(A)\,\textrm(B) = \exp\left(\tfrac\mathrm\log(A^\textB)\right)$
and on the observation that $\textrm(\sigma_y\otimes I_n)=(-i)^$.

Since calculating the logarithm of a matrix is a computationally demanding task, one can instead compute all eigenvalues of $((\sigma_y\otimes I_n)^T\cdot A)$, take the log of all of these and sum them up. This procedure merely exploits the property $\operatorname = \operatorname + \operatorname$. This can be implemented in Mathematica within a single line:

Pf _:= Module I^(n^2) Exp[ 1/2 Total[ Log[Eigenvalues[ Dot[Transpose[KroneckerProduct[PauliMatrix[2 IdentityMatrix[n], x] ]

However, this algorithm is unstable when the Pfaffian is large. The eigenvalues of $(\sigma_y\otimes I_n)^T\cdot A$ will generally be complex, and the logarithm of these complex eigenvalues are generally taken to be in \pi, \pi. Under the summation, for a real valued Pfaffian, the argument of the exponential will be given in the form $x + k\pi/2$ for some integer $k$. When $x$ is very large, rounding errors in computing the resulting sign from the complex phase can lead to a non-zero imaginary component.

For other (more) efficient algorithms see .

Applications

*There exist programs for the numerical computation of the Pfaffian on various platforms (Python, Matlab, Mathematica) .
*The Pfaffian is an invariant polynomial of a skew-symmetric matrix under a proper orthogonal change of basis. 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 is given by a Pfaffian, hence is polynomial time computable via the FKT algorithm. This is surprising given that for general graphs, the problem is very difficult (so called #P-complete). This result is used to calculate the number of domino tilings of a rectangle, the partition function of Ising models in physics, or of Markov random fields in machine learning (; ), where the underlying graph is planar. 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. Read Holographic algorithm for more information.

See also

*Determinant
*Dimer model
*Hafnian
*Polyomino
*Statistical mechanics

Notes

References

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* Reprinted in Collected mathematical papers, volume 2.
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External links

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* T. Jones
''The Pfaffian and the Wedge Product'' (a demonstration of the proof of the Pfaffian/determinant relationship)
* R. Kenyon and A. Okounkov
''What is ... a dimer?''
* {{OEIS el, 1=A004003, 2=Number of domino tilings (or dimer coverings), formalname=Number of domino tilings (or dimer coverings) of a 2n X 2n square
* W. Ledermann "A note on skew-symmetric determinants" https://www.researchgate.net/publication/231827602_A_note_on_skew-symmetric_determinants

Determinants
Multilinear algebra