An Analogue of Birkhoff's Problem III for Infinite Markov Matrices1

1969 ◽  
Vol 12 (5) ◽  
pp. 625-633
Author(s):  
Choo-Whan Kim
Keyword(s):  
Convex Hull ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Doubly Stochastic Matrices

A celebrated theorem of Birkhoff ([1], [6]) states that the set of n × n doubly stochastic matrices is identical with the convex hull of the set of n × n permutation matrices. Birkhoff [2, p. 266] proposed the problem of extending his theorem to the set of infinite doubly stochastic matrices. This problem, which is often known as Birkhoffs Problem III, was solved by Isbell ([3], [4]), Rattray and Peck [7], Kendall [5] and Révész [8].

scholarly journals On an Algorithm of G. Birkhoff Concerning Doubly Stochastic Matrices

1960 ◽  
Vol 3 (3) ◽  
pp. 237-242 ◽  
Author(s):  
Diane M. Johnson ◽  
A. L. Dulmage ◽  
N. S. Mendelsohn
Keyword(s):  
Convex Hull ◽  
Upper Bound ◽  
Stochastic Matrix ◽  
Stochastic Matrices ◽  
Von Neumann ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Doubly Stochastic Matrices ◽  
Set Of Points

In [1] G. Birkhoff stated an algorithm for expressing a doubly stochastic matrix as an average of permutation matrices. In this note we prove two graphical lemmas and use these to find an upper bound for the number of permutation matrices which the Birkhoff algorithm may use.A doubly stochastic matrix is a matrix of non-negative elements with row and column sums equal to unity and is there - fore a square matrix. A permutation matrix is an n × n doubly stochastic matrix which has n2-n zeros and consequently has n ones, one in each row and one in each column. It has been shown by Birkhoff [1],Hoffman and Wielandt [5] and von Neumann [7] that the set of all doubly stochastic matrices, considered as a set of points in a space of n2 dimensions constitute the convex hull of permutation matrices.

Notes on the Birkhoff Algorithm for Doubly Stochastic Matrices

1982 ◽  
Vol 25 (2) ◽  
pp. 191-199 ◽  
Author(s):  
Richard A. Brualdi
Keyword(s):  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Doubly Stochastic Matrices

AbstractThe purpose of this note is to tie together some results concerning doubly stochastic matrices and their representations as convex combinations of permutation matrices.

scholarly journals Linear maps leaving invariant subsets of nonnegative symmetric matrices

2003 ◽  
Vol 68 (2) ◽  
pp. 221-231 ◽  
Author(s):  
Hanley Chiang ◽  
Chi-Kwong Li
Keyword(s):  
Linear Transformation ◽  
Permutation Matrix ◽  
Symmetric Matrices ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Linear Maps ◽  
Doubly Stochastic Matrices ◽  
Low Dimensional

Let  be a certain set of nonnegative symmetric matrices, such as the set of symmetric doubly stochastic matrices or the set, of symmetric permutation matrices. It is proven that a linear transformation mapping  onto  must be of the form X ↦ PtX P for some permutation matrix P except for several low dimensional cases.

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

1979 ◽  
Vol 22 (1) ◽  
pp. 81-86 ◽  
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.

scholarly journals The Alternating Sign Matrix Polytope

10.37236/130 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Jessica Striker
Keyword(s):  
Convex Hull ◽  
Complete Characterization ◽  
Stochastic Matrices ◽  
Von Neumann ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Alternating Sign Matrix ◽  
Sign Matrix ◽  
Equivalent Description

We define the alternating sign matrix polytope as the convex hull of $n\times n$ alternating sign matrices and prove its equivalent description in terms of inequalities. This is analogous to the well known result of Birkhoff and von Neumann that the convex hull of the permutation matrices equals the set of all nonnegative doubly stochastic matrices. We count the facets and vertices of the alternating sign matrix polytope and describe its projection to the permutohedron as well as give a complete characterization of its face lattice in terms of modified square ice configurations. Furthermore we prove that the dimension of any face can be easily determined from this characterization.

Permanent Preservers on the Space of Doubly Stochastic Matrices

1962 ◽  
Vol 14 ◽  
pp. 190-194 ◽  
Author(s):  
B. N. Moyls ◽  
Marvin Marcus ◽  
Henryk Minc
Keyword(s):  
Linear Space ◽  
Stochastic Matrices ◽  
Linear Transformations ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Doubly Stochastic Matrices ◽  
Image Position

Let Mn be the linear space of n-square matrices with real elements. For a matrix X = (xij) ∈ Mn the permanent is defined bywhere σ runs over all permutations of 1, 2, …, n. In (2) Marcus and May determine the nature of all linear transformations T of Mn into itself such that per T(X) = per X for all X ∈ Mn. For such a permanent preserver T, and for n < 3, there exist permutation matrices P, Q, and diagonal matrices D, L in Mn, such that per DL = 1 and eitherorHere X′ denotes the transpose of X. In the case n = 2, a different type of transformation is also possible.

A short proof of the Farahat-Mirsky refinement of Birkhoff's theorem on doubly-stochastic matrices

1961 ◽  
Vol 57 (3) ◽  
pp. 681-681 ◽  
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.

scholarly journals The semigroup of doubly-stochastic matrices

1966 ◽  
Vol 7 (4) ◽  
pp. 178-183 ◽  
Author(s):  
H. K. Farahat
Keyword(s):  
Stochastic Matrix ◽  
Matrix Multiplication ◽  
Symmetric Groups ◽  
Maximal Subgroups ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Yield Information ◽  
Doubly Stochastic Matrices

The set Dn of all n × n doubly-stochastic matrices is a semigroup with respect to ordinary matrix multiplication. This note is concerned with the determination of the maximal subgroups of Dn. It is shown that the number of subgroups is finite, that each subgroup is finite and is in fact isomorphic to a direct product of symmetric groups. These results are applied in § 3 to yield information about the least number of permutation matrices whose convex hull contains a given doubly-stochastic matrix.

scholarly journals Positional Voting and Doubly Stochastic Matrices

2021 ◽  
Vol 128 (4) ◽  
pp. 337-351
Author(s):  
Jacqueline Anderson ◽  
Brian Camara ◽  
John Pike
Keyword(s):  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices ◽  
Positional Voting
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