Doubly stochastic matrix

The class of ${\displaystyle n\times n}$ doubly stochastic matrices is a convex polytope known as the Birkhoff polytope ${\displaystyle B_{n}}$. Using the matrix entries as Cartesian coordinates, it lies in an ${\displaystyle (n-1)^{2}}$-dimensional affine subspace of ${\displaystyle n^{2}}$-dimensional Euclidean space defined by ${\displaystyle 2n-1}$ independent linear constraints specifying that the row and column sums all equal one. (There are ${\displaystyle 2n-1}$ constraints rather than ${\displaystyle 2n}$ 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 ${\displaystyle B_{n}}$ is the convex hull of the set of ${\displaystyle n\times n}$ permutation matrices, and furthermore that the vertices of ${\displaystyle B_{n}}$ are precisely the permutation matrices. In other words, if ${\displaystyle A}$ is doubly stochastic matrix, then there exist ${\displaystyle \theta _{1},\ldots ,\theta _{k}\geq 0,\sum _{i=1}^{k}\theta _{i}=1}$ and permutation matrices ${\displaystyle P_{1},\ldots ,P_{k}}$ such that

${\displaystyle A=\theta _{1}P_{1}+\cdots +\theta _{k}P_{k}.}$

This representation is known as the Birkhoff–von Neumann decomposition, and it may not be unique in general. By Carathéodory's theorem, however, there exists a decomposition with no more than ${\displaystyle (n-1)^{2}+1=n^{2}-2n+2}$ terms, i.e. with [3]

${\displaystyle k\leq n^{2}-2n+2.}$

In other words, while there exists a decomposition with ${\displaystyle n!}$ permutation matrices, there is at least one constructible decomposition with no more than ${\displaystyle (n-1)^{2}+1}$ 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]

• The product of two doubly stochastic matrices is doubly stochastic. However, the inverse of a nonsingular doubly stochastic matrix need not be doubly stochastic (indeed, the inverse is doubly stochastic iff it has nonnegative entries).
• The stationary distribution of an 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 ${\displaystyle n=2}$, all bistochastic matrices are unistochastic and orthostochastic, but for larger ${\displaystyle n}$ this is not the case.
• Van der Waerden conjectured that the minimum permanent among all n × n doubly stochastic matrices is ${\displaystyle n!/n^{n}}$, achieved by the matrix for which all entries are equal to ${\displaystyle 1/n}$.[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]

• Stochastic matrix

• Brualdi, Richard A. (2006). Combinatorial matrix classes. Encyclopedia of Mathematics and Its Applications. 108. Cambridge: Cambridge University Press. ISBN 978-0-521-86565-4. Zbl 1106.05001.

• PlanetMath page on Birkhoff–von Neumann theorem
• PlanetMath page on proof of Birkhoff–von Neumann theorem