- Birkhoff polytope and Birkhoff–von Neumann theorem
- Other properties
- See also
- References
- External links
In mathematics, especially in probability and combinatorics, a doubly stochastic matrix
(also called bistochastic),
is a square matrix 
of nonnegative real numbers, each of whose rows and columns sums to 1,[1] i.e.,
,
Thus, a doubly stochastic matrix is both left stochastic and right stochastic.[1][2]
Indeed, any matrix that is both left and right stochastic must be square: 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.[1]
Birkhoff polytope and Birkhoff–von Neumann theorem
The class of 
doubly stochastic matrices is a convex polytope known as the Birkhoff polytope 
. Using the matrix entries as Cartesian coordinates, it lies in an 
-dimensional affine subspace of 
-dimensional Euclidean space defined by 
independent linear constraints specifying that the row and column sums all equal one. (There are constraints rather than 
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.
The Birkhoff–von Neumann theorem states that this polytope is the convex hull of the set of permutation matrices, and furthermore that the vertices of are precisely the permutation matrices. In other words, if 
is doubly stochastic matrix, then there exist 
and permutation matrices 
such that
This representation is known as the Birkhoff–von Neumann decomposition, and it may not be unique in general. By Marcus-Ree theorem, however, there need not be more than 
terms in any decomposition, namely[3]
In other words, while there exists a decomposition with 
permutation matrices, there is at least one constructible decomposition with no more than 
matrices.
The problem of computing the representation with the minimum number of terms has been shown to be NP-hard, but some heuristics for computing it are known.[4][5] This theorem can be extended for the general stochastic matrix with deterministic transition matrices.[6]
Other properties
- The inverse of a nonsingular doubly stochastic matrix need not be doubly stochastic.
- The stationary distribution of a irreducible aperiodic finite Markov chain is uniform if and only if its transition matrix is doubly stochastic.
- Sinkhorn's theorem states that any matrix with strictly positive entries can be made doubly stochastic by pre- and post-multiplication by diagonal matrices.
- For
, all bistochastic matrices are unistochastic and orthostochastic, but for larger 
it is not the case. - Van der Waerden conjectured that the minimum permanent among all {{nowrap|n × n}} doubly stochastic matrices is
, achieved by the matrix for which all entries are equal to 
.[7] Proofs of this conjecture were published in 1980 by B. Gyires[8] and in 1981 by G. P. Egorychev[9] and D. I. Falikman;[10] for this work, Egorychev and Falikman won the Fulkerson Prize in 1982.[11]
See also
- Stochastic matrix
References
2. ^{{cite book|last=Marshal, Olkin|title=Inequalities: Theory of Majorization and Its Applications|year=1979|isbn=978-0-12-473750-1|pages=8}}
3. ^{{cite journal|last1=Marcus|first1=M.|last2=Ree|first2=R.|title=Diagonals of doubly stochastic matrices|journal=The Quarterly Journal of Mathematics|date=1959|volume=10|issue=1|pages=296–302|doi=10.1093/qmath/10.1.296}}
4. ^{{cite journal|last1=Dufossé|first1=Fanny|last2=Uçar|first2=Bora|title=Notes on Birkhoff–von Neumann decomposition of doubly stochastic matrices|journal=Linear Algebra and its Applications|date=May 2016|volume=497|pages=108–115|doi=10.1016/j.laa.2016.02.023}}
5. ^{{Cite journal|last=Dufossé|first=Fanny|last2=Kaya|first2=Kamer|last3=Panagiotas|first3=Ioannis|last4=Uçar|first4=Bora|date=2018|title=Further notes on Birkhoff–von Neumann decomposition of doubly stochastic matrices|journal=Linear Algebra and its Applications|volume=554|pages=68–78|doi=10.1016/j.laa.2018.05.017|issn=0024-3795}}
6. ^{{Cite journal|last=Ye|first=Felix X.-F.|last2=Wang|first2=Yue|last3=Qian|first3=Hong|title=Stochastic dynamics: Markov chains and random transformations|journal=Discrete and Continuous Dynamical Systems - Series B|volume=21|issue=7|pages=2337–2361|doi=10.3934/dcdsb.2016050|year=2016}}
7. ^{{citation | last = van der Waerden | first = B. L. | author-link = Bartel Leendert van der Waerden | journal = Jber. Deutsch. Math.-Verein. | page = 117 | title = Aufgabe 45 | volume = 35 | year = 1926}}.
8. ^{{citation | last = Gyires | first = B. | issue = 3–4 | journal = Publicationes Mathematicae Institutum Mathematicum Universitatis Debreceniensis | mr = 604006 | pages = 291–304 | title = The common source of several inequalities concerning doubly stochastic matrices | volume = 27 | year = 1980}}.
9. ^{{citation | last = Egoryčev | first = G. P. | language = Russian | location = Krasnoyarsk | mr = 602332 | page = 12 | publisher = Akad. Nauk SSSR Sibirsk. Otdel. Inst. Fiz. | title = Reshenie problemy van-der-Vardena dlya permanentov | year = 1980}}. {{citation | last = Egorychev | first = G. P. | issue = 6 | journal = Akademiya Nauk SSSR | language = Russian | mr = 638007 | pages = 65–71, 225 | title = Proof of the van der Waerden conjecture for permanents | volume = 22 | year = 1981}}. {{citation | last = Egorychev | first = G. P. | doi = 10.1016/0001-8708(81)90044-X | issue = 3 | journal = Advances in Mathematics | mr = 642395 | pages = 299–305 | title = The solution of van der Waerden's problem for permanents | volume = 42 | year = 1981}}.
10. ^{{citation | last = Falikman | first = D. I. | issue = 6 | journal = Akademiya Nauk Soyuza SSR | language = Russian | mr = 625097 | pages = 931–938, 957 | title = Proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix | volume = 29 | year = 1981}}.
11. ^Fulkerson Prize, Mathematical Optimization Society, retrieved 2012-08-19.
- {{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=978-0-521-86565-4 | zbl=1106.05001 }}
External links
- PlanetMath page on Birkhoff–von Neumann theorem
- PlanetMath page on proof of Birkhoff–von Neumann theorem
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