A variational proof of Birkhoff's theorem on doubly stochastic matrices

1961 ◽

Vol 57 (3) ◽

pp. 681-681 ◽

Cited By ~ 4

Author(s):

J. M. Hammersley

Keyword(s):

Short Proof ◽

Convex Combination ◽

Stochastic Matrix ◽

Stochastic Matrices ◽

Doubly Stochastic Matrix ◽

Doubly Stochastic ◽

Permutation Matrices ◽

Doubly Stochastic Matrices ◽

Birkhoff's Theorem ◽

Birkhoff’S Theorem

A doubly-stochastic matrix is an n × n matrix with non-negative elements such that each row and each column sums to 1. A permutation matrix is the result of permuting the rows of the unit n × n matrix. Birkhoff's theorem states that the doubly-stochastic matrices constitute the convex hull of the permutation matrices. Using Birkhoff's theorem, Farahat and Mirsky (1) showed that each doubly-stochastic matrix could be expressed as a convex combination of n2 − 2n + 2 permutation matrices, though not in general of fewer. Given Birkhoff's theorem, the Farahat-Mirsky refinement can also be proved quite shortly as follows.

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A short note on extreme points of certain polytopes

Special Matrices ◽

10.1515/spma-2020-0005 ◽

2020 ◽

Vol 8 (1) ◽

pp. 36-39

Author(s):

Lei Cao ◽

Ariana Hall ◽

Selcuk Koyuncu

Keyword(s):

Short Proof ◽

Convex Polytopes ◽

Short Note ◽

Convex Polytope ◽

Extreme Points ◽

Alternative Proof ◽

Stochastic Matrices ◽

Doubly Stochastic ◽

Birkhoff's Theorem ◽

Birkhoff’S Theorem

AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

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An Elementary Proof of Johnson–Dulmage–Mendelsohn's Refinement of Birkhoff's Theorem on Doubly Stochastic Matrices

Canadian Mathematical Bulletin ◽

10.4153/cmb-1979-011-4 ◽

1979 ◽

Vol 22 (1) ◽

pp. 81-86 ◽

Cited By ~ 2

Author(s):

Akihiro Nishi

Keyword(s):

Upper Bound ◽

Elementary Proof ◽

Convex Combination ◽

Stochastic Matrix ◽

Stochastic Matrices ◽

Doubly Stochastic Matrix ◽

Doubly Stochastic ◽

Permutation Matrices ◽

Doubly Stochastic Matrices ◽

Birkhoff’S Theorem

SummaryA purely combinatorial and elementary proof of Johnson-Dulmage-Mendelsohn's theorem, which gives a quite sharp upper bound on the number of permutation matrices needed for representing a doubly stochastic matrix by their convex combination, is given.

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