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| {{distinguish|Pfaffian function|Pfaffian system}}
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| 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 term ''Pfaffian'' was introduced by {{harvs|txt|authorlink=Arthur Cayley|last=Cayley|year=1852}} who named them after [[Johann Friedrich Pfaff]]. The Pfaffian is nonvanishing only for 2''n'' × 2''n'' skew-symmetric matrices, in which case it is a polynomial of degree ''n''.
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| Explicitly, for a skew-symmetric matrix '''A''',
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| :<math> \operatorname{pf(A)}^2=\det(A),</math>
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| which was first proved by [[Thomas Muir (mathematician)|Thomas Muir]] in 1882 {{harv|Muir|1882}}.
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| The fact that the determinant of any skew symmetric matrix is the square of a polynomial can be shown by writing the matrix as a block matrix,
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| then using induction and examining the [[Schur complement]], which is skew symmetric as well.
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| <ref>Ledermann, W. "A note on skew-symmetric determinants"</ref>
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| ==Examples==
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| :<math>A=\begin{bmatrix} 0 & a \\ -a & 0 \end{bmatrix}.\qquad\operatorname{pf(A)}=a.</math> | |
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| :<math>B=\begin{bmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{bmatrix}.\qquad\operatorname{pf(B)}=0.</math>
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| (3 is odd, so Pfaffian of B is 0)
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| :<math>\operatorname{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>
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| The Pfaffian of a 2''n'' × 2''n'' skew-symmetric tridiagonal matrix is given as
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| :<math>\operatorname{pf}\begin{bmatrix}
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| 0 & a_1\\ -a_1 & 0 & b_1\\ 0 & -b_1 &0 & a_2 \\ 0 & 0 & -a_2 &\ddots&\ddots\\
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| &&&\ddots&&b_{n-1}\\
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| &&&&-b_{n-1}&0&a_n\\
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| &&&&&-a_n&0
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| \end{bmatrix} = a_1 a_2\cdots a_n.</math>
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| which contains the important case of a 2''n'' × 2''n'' skew-symmetric matrix with 2 × 2 blocks on the
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| diagonal:
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| :<math>\operatorname{pf}\begin{bmatrix}
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| \begin{matrix} 0 & \lambda_1\\ -\lambda_1 & 0\end{matrix} & 0 & \cdots & 0 \\
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| 0 & \begin{matrix}0 & \lambda_2\\ -\lambda_2 & 0\end{matrix} & & 0 \\
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| \vdots & & \ddots & \vdots \\
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| 0 & 0 & \cdots & \begin{matrix}0 & \lambda_n\\ -\lambda_n & 0\end{matrix}
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| \end{bmatrix} = \lambda_1\lambda_2\cdots\lambda_n.</math>
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| (Note that any skew-symmetric matrix can be reduced to this form, see [[Skew-symmetric_matrix#Spectral_theory|Spectral theory of a skew-symmetric matrix]])
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| ==Formal definition==
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| Let ''A'' = {''a''<sub>''i,j''</sub>} be a 2''n'' × 2''n'' skew-symmetric matrix. The Pfaffian of ''A'' is defined by the equation
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| :<math>\operatorname{pf}(A) = \frac{1}{2^n n!}\sum_{\sigma\in S_{2n}}\operatorname{sgn}(\sigma)\prod_{i=1}^{n}a_{\sigma(2i-1),\sigma(2i)}</math>
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| where ''S''<sub>2''n''</sub> is the [[symmetric group]] and sgn(σ) is the [[signature (permutation)|signature]] of σ.
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| One can make use of the skew-symmetry of ''A'' to avoid summing over all possible [[permutation]]s. Let Π be the set of all [[partition of a set|partition]]s of {1, 2, …, 2''n''} into pairs without regard to order. There are (2''n'' − 1)[[double factorial|!!]] such partitions. An element α ∈ Π can be written as
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| :<math>\alpha=\{(i_1,j_1),(i_2,j_2),\cdots,(i_n,j_n)\}</math> | |
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| with ''i''<sub>''k''</sub> < ''j''<sub>''k''</sub> and <math>i_1 < i_2 < \cdots < i_n</math>. Let
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| :<math>\pi=\begin{bmatrix} 1 & 2 & 3 & 4 & \cdots & 2n \\ i_1 & j_1 & i_2 & j_2 & \cdots & j_{n} \end{bmatrix}</math>
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| be the corresponding permutation. Given a partition α as above, define
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| :<math> A_\alpha =\operatorname{sgn}(\pi)a_{i_1,j_1}a_{i_2,j_2}\cdots a_{i_n,j_n}.</math>
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| The Pfaffian of ''A'' is then given by
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| :<math>\operatorname{pf}(A)=\sum_{\alpha\in\Pi} A_\alpha.</math>
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| 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, <math>\det\,A = \det\,A^\text{T} = \det\left(-A\right) = (-1)^n \det\,A</math>, and for ''n'' odd, this implies <math>\det\,A = 0</math>.
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| ===Recursive definition===
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| 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
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| :<math>\operatorname{pf}(A)=\sum_{i=2}^{2n}(-1)^{i}a_{1i}\operatorname{pf}(A_{\hat{1}\hat{i}}),</math>
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| where <math>A_{\hat{1}\hat{i}}</math> denotes the matrix ''A'' with both the first and ''i''-th rows and columns removed.
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| ===Alternative definitions===
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| * One can associate to any skew-symmetric 2''n''×2''n'' matrix ''A'' ={''a''<sub>''ij''</sub>} a [[exterior algebra|bivector]]
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| :<math>\omega=\sum_{i<j} a_{ij}\;e^i\wedge e^j.</math>
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| where {''e''<sup>1</sup>, ''e''<sup>2</sup>, …, ''e''<sup>2''n''</sup>} is the standard basis of '''R'''<sup>2n</sup>. The Pfaffian is then defined by the equation
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| :<math>\frac{1}{n!}\omega^n = \operatorname{pf}(A)\;e^1\wedge e^2\wedge\cdots\wedge e^{2n},</math>
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| here ω<sup>''n''</sup> denotes the [[wedge product]] of ''n'' copies of ω.
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| ==Identities==
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| For a 2''n'' × 2''n'' skew-symmetric matrix ''A''
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| :<math>\operatorname{pf}(A^\text{T}) = (-1)^n\operatorname{pf}(A).</math>
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| :<math>\operatorname{pf}(\lambda A) = \lambda^n \operatorname{pf}(A).</math>
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| :<math>\operatorname{pf}(A)^2 = \det(A).</math>
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| For an arbitrary 2''n'' × 2''n'' matrix ''B'',
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| :<math>\operatorname{pf}(BAB^\text{T})= \det(B)\operatorname{pf}(A).</math>
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| Substituting in this equation ''B = A<sup>m</sup>'', one gets for all integer ''m''
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| :<math>\operatorname{pf}(A^{2m+1})= (-1)^{nm}\operatorname{pf}(A)^{2m+1}.</math>
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| For a block-diagonal matrix
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| ::<math>A_1\oplus A_2=\begin{bmatrix} A_1 & 0 \\ 0 & A_2 \end{bmatrix},</math>
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| :<math>\operatorname{pf}(A_1\oplus A_2) =\operatorname{pf}(A_1)\operatorname{pf}(A_2).</math>
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| For an arbitrary ''n'' × ''n'' matrix ''M'':
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| :<math>\operatorname{pf}\begin{bmatrix} 0 & M \\ -M^\text{T} & 0 \end{bmatrix} =
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| (-1)^{n(n-1)/2}\det M.</math>
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| If ''A'' depends on some variable ''x''<sub>''i''</sub>, then the gradient of a Pfaffian is given by
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| :<math>\frac{1}{\operatorname{pf}(A)}\frac{\partial\operatorname{pf}(A)}{\partial x_i}=\frac{1}{2}\operatorname{tr}\left(A^{-1}\frac{\partial A}{\partial x_i}\right),</math>
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| and the [[Hessian matrix|Hessian]] of a Pfaffian is given by
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| :<math>\frac{1}{\operatorname{pf}(A)}\frac{\partial^2\operatorname{pf}(A)}{\partial x_i\partial x_j}=\frac{1}{2}\operatorname{tr}\left(A^{-1}\frac{\partial^2 A}{\partial x_i\partial x_j}\right)-\frac{1}{2}\operatorname{tr}\left(A^{-1}\frac{\partial A}{\partial x_i}A^{-1}\frac{\partial A}{\partial x_j}\right)+\frac{1}{4}\operatorname{tr}\left(A^{-1}\frac{\partial A}{\partial x_i}\right)\operatorname{tr}\left(A^{-1}\frac{\partial A}{\partial x_j}\right).</math>
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| ==Properties==
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| Pfaffians have the following properties, which are similar to those of determinants.
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| * Multiplication of a row and a column by a constant is equivalent to multiplication of Pfaffian by the same constant.
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| * Simultaneous interchange of two different rows and corresponding columns changes the sign of Pfaffian.
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| * A multiple of a row and corresponding column added to another row and corresponding column does not change the value of Pfaffian.
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| These properties can be derived from the identity <math>\operatorname{pf}(BAB^\text{T})=\det(B)\operatorname{pf}(A)</math>.
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| ==Applications==
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| *There exist programs for the numerical computation of the Pfaffian on various platforms (Python, Matlab, Mathematica) {{harv|Wimmer|2012}}.
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| *The Pfaffian is an [[invariant polynomial]] of a skew-symmetric matrix under a proper [[orthogonal group|orthogonal]] change of basis. As such, it is important in the theory of [[characteristic class]]es. In particular, it can be used to define the [[Euler class]] of a [[Riemannian manifold]] which is used in the [[generalized Gauss–Bonnet theorem]].
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| *The number of [[perfect matching]]s 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 [[Sharp-P-complete|#P-complete]]). This result is used to calculate the number of [[domino tiling]]s of a rectangle, the [[partition function (statistical mechanics)|partition function]] of [[Ising model]]s in physics, or of [[Markov random fields]] in [[machine learning]] ({{harvnb|Globerson|Jaakkola|2007}}; {{harvnb|Schraudolph|Kamenetsky|2009}}), 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.
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| ==See also==
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| *[[Determinant]]
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| *[[Dimer model]]
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| *[[Polyomino]]
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| *[[Statistical mechanics]]
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| == References ==
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| {{Reflist}}
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| *{{Cite journal | last1=Cayley | first1=Arthur | author1-link=Arthur Cayley | title=On the theory of permutants | url=http://www.archive.org/stream/collectedmathema02cayluoft#page/16/mode/2up | year=1852 | journal=Cambridge and Dublin Mathematical Journal | volume=VII | pages=40–51 | ref=harv | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}} Reprinted in Collected mathematical papers, volume 2.
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| * {{cite journal | ref=harv
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| | title = The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice
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| | journal = Physica
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| | volume = 27 | issue = 12 | year = 1961 | pages = 1209–1225
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| | first = P. W. | last = Kasteleyn
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| | authorlink = Pieter_Kasteleyn
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| | doi = 10.1016/0031-8914(61)90063-5}}
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| * {{cite arXiv | first=James | last=Propp | year = 2004 | title="Lambda-determinants and domino-tilings" | eprint=math/0406301 | eprint=math.CO/0406301 | ref=harv | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite arxiv to end in a ".", as necessary. -->{{inconsistent cite arxivs}}}}.
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| * {{Cite journal | first=Amir | last=Globerson | first2=Tommi | last2=Jaakkola | year = 2007 | contribution= ''Approximate inference using planar graph decomposition'' | contribution-url= http://books.nips.cc/papers/files/nips19/NIPS2006_0703.pdf | title=''Advances in Neural Information Processing Systems 19'' | url=http://books.nips.cc/ | publisher=MIT Press | ref=harv | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}.
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| * {{Cite journal | first=Nicol | last=Schraudolph | first2=Dmitry | last2=Kamenetsky | year = 2009 | contribution= ''Efficient exact inference in planar Ising models'' | contribution-url= http://books.nips.cc/papers/files/nips21/NIPS2008_0401.pdf | title=''Advances in Neural Information Processing Systems 21'' | url=http://books.nips.cc/ | publisher=MIT Press | ref=harv | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}.
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| * {{cite journal | ref=harv | journal=The Games and Puzzles Journal | volume=2 | year=1996 | pages=204–5 | first1=G.P. |last=Jeliss | first2=Robin J. | last2=Chapman | title=Dominizing the Chessboard | issue=14}}
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| * {{cite journal | ref=harv | title=Domino Tilings and Products of Fibonacci and Pell numbers| journal=Journal of Integer Sequences | volume=5 | year=2002 | first=James A. | last=Sellers | issue=1 |page=02.1.2 | url=http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Sellers/sellers4.html }}
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| * {{cite book | ref=harv | title=[[The Penguin Dictionary of Curious and Interesting Numbers]] | edition=revised | year=1997 | isbn=0-14-026149-4 | first=David | last=Wells | page=182 }}
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| * {{cite book | ref=harv | title=A Treatise on the Theory of Determinants | year=1882 | publisher=Macmillan and Co. | first=Thomas | last=Muir}} [http://books.google.com/books?id=pk4DAAAAQAAJ&dq=thomas+muir+treatise+on+the+theory+of+determinant&psp=1 ''Online'']
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| * {{cite journal | ref=harv | title=Skew-Symmetric Determinants | journal=The American Mathematical Monthly | volume=61 | year=1954 | page=116 | first=S. | last=Parameswaran | jstor=2307800 | issue=2}}
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| * {{cite journal| ref=harv |title=Efficient numerical computation of the Pfaffian for dense and banded skew-symmetric matrices|journal=[[ACM Trans. Math. Software]]|volume=38|page=30|year=2012 | first=M. | last=Wimmer | arxiv=1102.3440}}
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| ==External links==
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| * [http://planetmath.org/encyclopedia/Pfaffian.html Pfaffian at PlanetMath.org]
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| * T. Jones, [http://www.physics.drexel.edu/~tim/open/pfaff/pfaff.pdf ''The Pfaffian and the Wedge Product'' (a demonstration of the proof of the Pfaffian/determinant relationship)]
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| * R. Kenyon and [[Andrei Okounkov|A. Okounkov]], [http://www.ams.org/notices/200503/what-is.pdf ''What is ... a dimer?'']
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| * {{OEIS|id=A004003}}
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| * W. Ledermann "A note on skew-symmetric determinants" http://www.researchgate.net/publication/231827602_A_note_on_skew-symmetric_determinants
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| [[Category:Determinants]]
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| [[Category:Multilinear algebra]]
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