Inequalities for permanents and permanental minors

1965 ◽  
Vol 61 (3) ◽  
pp. 741-746 ◽  
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.

Permanents of Random Doubly Stochastic Matrices

1974 ◽  
Vol 26 (3) ◽  
pp. 600-607 ◽  
Author(s):  
R. C. Griffiths
Keyword(s):  
Symmetric Group ◽  
Stochastic Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Survey Article ◽  
Doubly Stochastic Matrices ◽  
Image Position

The permanent of an n × n matrix A = (aij) is defined aswhere Sn is the symmetric group of order n. For a survey article on permanents the reader is referred to [2]. An unresolved conjecture due to van der Waerden states that if A is an n × n doubly stochastic matrix; then per (A) ≧ n!/nn, with equality if and only if A = Jn = (1/n).

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.

Extreme points of convex sets of doubly stochastic matrices. II

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.

Infinite Doubly Stochastic Matrices

1962 ◽  
Vol 5 (1) ◽  
pp. 1-4 ◽  
Author(s):  
J.R. Isbell
Keyword(s):  
Linear Space ◽  
Convex Set ◽  
The Other ◽  
Locally Convex Spaces ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Incorrect Assertion ◽  
Doubly Stochastic Matrices ◽  
The Axiom Of Choice ◽  
Locally Convex

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.

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.

On the Permanent of a Doubly Stochastic Matrix

1966 ◽  
Vol 18 ◽  
pp. 758-761 ◽  
Author(s):  
Herbert S. Wilf
Keyword(s):  
Stochastic Matrix ◽  
Upper Bounds ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices ◽  
Image Position

If is an n X n matrix, the permanent of A, Per A, is defined by1where the sum is over all permutations. If A is doubly stochastic (i.e., nonnegative with row and column sums all equal to 1), then it has been conjectured that Per A ⩾ n!/nn. When confronted with a vaguely similar problem about determinants, M. Kac (1) observed that the computation of minima can often be aided by knowledge of various averages. In this spirit we calculate here the average permanent of a class of doubly stochastic matrices and thereby obtain upper bounds for the minima. These turn out to be surprisingly sharp.

On a conjecture of B. L. van der Waerden

1967 ◽  
Vol 63 (2) ◽  
pp. 305-309 ◽  
Author(s):  
Marvin Marcus ◽  
Henryk Minc
Keyword(s):  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices ◽  
Image Position ◽  
Square Matrix

Let Ωn be the set of all n-square doubly stochastic matrices, i.e. matrices with non-negative entries all of whose row sums and column sums are 1. If A = (aij) is an n-square matrix let per (A) denote the permanent of A, that is.

An expansion for the permanent of a doubly stochastic matrix

1973 ◽  
Vol 15 (4) ◽  
pp. 504-509
Author(s):  
R. C. Griffiths
Keyword(s):  
Symmetric Group ◽  
Line Segment ◽  
Convex Set ◽  
Stochastic Matrix ◽  
Weighted Average ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Survey Article ◽  
The Matrix ◽  
Image Position

The permanent of an n-square matrix A = (aij) is defined by where Sn is the symmetric group of order n. Kn will denote the convex set of all n-square doubly stochastic matrices and K0n its interior. Jn ∈ Kn will be the matrix with all elements equal to 1/n. If M ∈ K0n, then M lies on a line segment passing through Jn and another B ∈ Kn — K0n. This note gives an expansion for the permanent of such a line segment as a weighted average of permanents of matrices in Kn. For a survey article on permanents the reader is referred to Marcus and Mine [3].

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