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.

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.

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].

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.

On the Extreme Rays of the Metric Cone

1980 ◽  
Vol 32 (1) ◽  
pp. 126-144 ◽  
Author(s):  
David Avis
Keyword(s):  
Convex Set ◽  
Convex Combination ◽  
Extreme Points ◽  
Convex Polyhedra ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Image Position ◽  
Polyhedral Convex Set ◽  
Extreme Rays

A classical result in the theory of convex polyhedra is that every bounded polyhedral convex set can be expressed either as the intersection of half-spaces or as a convex combination of extreme points. It is becoming increasingly apparent that a full understanding of a class of convex polyhedra requires the knowledge of both of these characterizations. Perhaps the earliest and neatest example of this is the class of doubly stochastic matrices. This polyhedron can be defined by the system of equationsBirkhoff [2] and Von Neuman have shown that the extreme points of this bounded polyhedron are just the n × n permutation matrices. The importance of this result for mathematical programming is that it tells us that the maximum of any linear form over P will occur for a permutation matrix X.

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.

scholarly journals Further notes on Birkhoff–von Neumann decomposition of doubly stochastic matrices

2018 ◽  
Vol 554 ◽  
pp. 68-78 ◽  
Author(s):  
Fanny Dufossé ◽  
Kamer Kaya ◽  
Ioannis Panagiotas ◽  
Bora Uçar
Keyword(s):  
Stochastic Matrices ◽  
Von Neumann ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices

scholarly journals Preconditioning Techniques Based on the Birkhoff–von Neumann Decomposition

2017 ◽  
Vol 17 (2) ◽  
pp. 201-215 ◽  
Author(s):  
Michele Benzi ◽  
Bora Uçar
Keyword(s):  
Linear Systems ◽  
Numerical Experiments ◽  
Sparse Matrices ◽  
Stochastic Matrices ◽  
Preconditioning Techniques ◽  
Von Neumann ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices ◽  
Theoretical Results

AbstractWe introduce a class of preconditioners for general sparse matrices based on the Birkhoff–von Neumann decomposition of doubly stochastic matrices. These preconditioners are aimed primarily at solving challenging linear systems with highly unstructured and indefinite coefficient matrices. We present some theoretical results and numerical experiments on linear systems from a variety of applications.

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 Spectrality of elementary operators

1990 ◽  
Vol 49 (2) ◽  
pp. 327-346 ◽  
Author(s):  
Milan Hladnik
Keyword(s):  
Complete Characterization ◽  
Linear Operators ◽  
Schatten Classes ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Normal Operators ◽  
Elementary Operators

AbstractSpectrality and prespectrality of elementary operators , acting on the algebra B(k) of all bounded linear operators on a separable infinite-dimensional complex Hubert space K, or on von Neumann-Schatten classes in B(k), are treated. In the case when (a1, a2, …, an) and (b1, b2, …, bn) are two n—tuples of commuting normal operators on H, the complete characterization of spectrality is given.

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