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| In [[mathematics]], especially in [[probability]] and [[combinatorics]], a '''doubly stochastic matrix'''
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| (also called bistochastic),
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| is a [[square matrix]] <math>A=(a_{ij})</math> of nonnegative [[real number]]s, each of whose rows and columns sum to 1, i.e.,
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| :<math>\sum_i a_{ij}=\sum_j a_{ij}=1</math>,
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| Thus, a doubly stochastic matrix is both left [[stochastic matrix|stochastic]] and right stochastic.<ref>{{cite book|last=Marshal, Olkin|title=Inequalities: Theory of Majorization and Its Applications|year=1979|isbn=0-12-473750-1|pages=8}}</ref>
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| Such a transition matrix is necessarily a [[square matrix]]: if every row sums to one then the sum of all entries in the matrix must be equal to the number of rows, and since the same holds for columns, the number of rows and columns must be equal.
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| ==Birkhoff polytope and Birkhoff–von Neumann theorem==
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| The class of <math>n\times n</math> doubly stochastic matrices is a [[convex polytope]] known as the [[Birkhoff polytope]] <math>B_n</math>. Using the matrix entries as [[Cartesian coordinates]], it lies in an <math>(n-1)^2</math>-dimensional affine subspace of <math>n^2</math>-dimensional [[Euclidean space]]. defined by <math>2n-1</math> independent linear constraints specifying that the row and column sums all equal one. (There are <math>2n-1</math> constraints rather than <math>2n</math> because one of these constraints is dependent, as the sum of the row sums must equal the sum of the column sums.) Moreover, the entries are all constrained to be non-negative and less than or equal to one.
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| The '''Birkhoff–von Neumann theorem''' states that this polytope <math>B_n</math> is the [[convex hull]] of the set of <math>n\times n</math> [[permutation matrix|permutation matrices]], and furthermore that the [[vertex (geometry)|vertices]] of <math>B_n</math> are precisely the permutation matrices.
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| ==Other properties==
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| [[Sinkhorn's theorem]] states that any matrix with strictly positive entries can be made doubly stochastic by pre- and post-multiplication by [[diagonal matrix|diagonal matrices]].
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| For <math>n=2</math>, all bistochastic matrices are [[unistochastic matrix|unistochastic]] and [[orthostochastic matrix|orthostochastic]], but for larger <math>n</math> it is not the case.
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| [[Bartel Leendert van der Waerden|Van der Waerden]] conjectured that the minimum [[permanent]] among all {{nowrap|''n'' × ''n''}} doubly stochastic matrices is <math>n!/n^n</math>, achieved by the matrix for which all entries are equal to <math>1/n</math>.<ref>{{citation | |
| | last = van der Waerden | first = B. L. | author-link = Bartel Leendert van der Waerden
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| | journal = Jber. Deutsch. Math.-Verein.
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| | page = 117
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| | title = Aufgabe 45
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| | volume = 35
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| | year = 1926}}.</ref> Proofs of this conjecture were published in 1980 by B. Gyires<ref>{{citation
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| | last = Gyires | first = B.
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| | issue = 3-4
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| | journal = Publicationes Mathematicae Institutum Mathematicum Universitatis Debreceniensis
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| | mr = 604006
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| | pages = 291–304
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| | title = The common source of several inequalities concerning doubly stochastic matrices
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| | volume = 27
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| | year = 1980}}.</ref> and in 1981 by G. P. Egorychev<ref>{{citation
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| | last = Egoryčev | first = G. P.
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| | language = Russian
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| | location = Krasnoyarsk
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| | mr = 602332
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| | page = 12
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| | publisher = Akad. Nauk SSSR Sibirsk. Otdel. Inst. Fiz.
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| | title = Reshenie problemy van-der-Vardena dlya permanentov
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| | year = 1980}}. {{citation
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| | last = Egorychev | first = G. P.
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| | issue = 6
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| | journal = Akademiya Nauk SSSR
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| | language = Russian
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| | mr = 638007
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| | pages = 65–71, 225
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| | title = Proof of the van der Waerden conjecture for permanents
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| | volume = 22
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| | year = 1981}}. {{citation
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| | last = Egorychev | first = G. P.
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| | doi = 10.1016/0001-8708(81)90044-X
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| | issue = 3
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| | journal = Advances in Mathematics
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| | mr = 642395
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| | pages = 299–305
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| | title = The solution of van der Waerden's problem for permanents
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| | volume = 42
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| | year = 1981}}.</ref> and D. I. Falikman;<ref>{{citation
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| | last = Falikman | first = D. I.
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| | issue = 6
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| | journal = Akademiya Nauk Soyuza SSR
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| | language = Russian
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| | mr = 625097
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| | pages = 931–938, 957
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| | title = Proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix
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| | volume = 29
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| | year = 1981}}.</ref> for this work, Egorychev and Falikman won the [[Fulkerson Prize]] in 1982.<ref>[http://www.mathopt.org/?nav=fulkerson Fulkerson Prize], Mathematical Optimization Society, retrieved 2012-08-19.</ref>
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| ==See also==
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| *[[Stochastic matrix]]
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| ==References==
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| {{Reflist}}
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| * {{cite book | last=Brualdi | first=Richard A. | title=Combinatorial matrix classes | series=Encyclopedia of Mathematics and Its Applications | volume=108 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86565-4 | zbl=1106.05001 }}
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| ==External links==
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| * [http://planetmath.org/birkhoffvonneumanntheorem PlanetMath page on Birkhoff–von Neumann theorem]
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| * [http://planetmath.org/proofofbirkhoffvonneumanntheorem PlanetMath page on proof of Birkhoff–von Neumann theorem]
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| [[Category:Matrices]]
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