Infinite Doubly Stochastic Matrices

This note proves two propositions on infinite doubly stochastic matrices, both of which already appear in the literature: one with an unnecessarily sophisticated proof (Kendall [2]) and the other with the incorrect assertion that the proof is trivial (Isbell [l]). Both are purely algebraic; so we are, if you like, in the linear space of all real doubly infinite matrices A = (aij).Proposition 1. Every extreme point of the convex set of ail doubly stochastic matrices is a permutation matrix.Kendall's proof of this depends on an ingenious choice of a topology and the Krein-Milman theorem for general locally convex spaces [2]. The following proof depends on practically nothing: for example, not on the axiom of choice.

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Inequalities for permanents and permanental minors

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100039086 ◽

1965 ◽

Vol 61 (3) ◽

pp. 741-746 ◽

Cited By ~ 5

Author(s):

R. A. Brualdi ◽

M. Newman

Keyword(s):

Convex Function ◽

Convex Set ◽

Identity Matrix ◽

Stochastic Matrices ◽

Doubly Stochastic ◽

Counter Example ◽

Doubly Stochastic Matrices ◽

Permanent Function ◽

Image Position

Let Ωndenote the convex set of alln×ndoubly stochastic matrices: chat is, the set of alln×nmatrices with non-negative entries and row and column sums 1. IfA= (aij) is an arbitraryn×nmatrix, then thepermanentofAis the scalar valued function ofAdefined bywhere the subscriptsi1,i2, …,inrun over all permutations of 1, 2, …,n. The permanent function has been studied extensively of late (see, for example, (1), (2), (3), (4), (6)) and it is known that ifA∈ Ωnthen 0 <cn≤ per (A) ≤ 1, where the constantcndepends only onn. It is natural to inquire if per (A) is a convex function ofAforA∈ Ωn. That this is not the case was shown by a counter-example given by Marcus and quoted by Perfect in her paper ((5)). In this paper, however, she shows that per (½I+ ½A) ≤ ½ + ½ per (A) for allA∈ Ωn. HereI=Inis the identity matrix of ordern.

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Extreme points of convex sets of doubly stochastic matrices. II

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100051768 ◽

1975 ◽

Vol 78 (2) ◽

pp. 327-331

Author(s):

J. G. Mauldon

Keyword(s):

Convex Sets ◽

Convex Set ◽

Extreme Points ◽

Stochastic Matrices ◽

Doubly Stochastic ◽

Doubly Stochastic Matrices

We prove a conjecture of (5), namely that the convex set of all infinite doubly stochastic matrices whose entries are all strictly less than θ(0 < θ ≤ 1) possesses extreme points if and only if θ is irrational.

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Permanent Preservers on the Space of Doubly Stochastic Matrices

Canadian Journal of Mathematics ◽

10.4153/cjm-1962-014-1 ◽

1962 ◽

Vol 14 ◽

pp. 190-194 ◽

Cited By ~ 4

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.

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