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.

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 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.

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.

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).

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.

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 Diagonal Sums of Doubly Substochastic Matrices

2019 ◽  
Vol 35 ◽  
pp. 42-52
Author(s):  
Lei Cao ◽  
Zhi Chen ◽  
Xuefeng Duan ◽  
Selcuk Koyuncu ◽  
Huilan Li
Keyword(s):  
Symmetric Group ◽  
Stochastic Matrix ◽  
Convex Polytope ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices

Let $\Omega_n$ denote the convex polytope of all $n\times n$ doubly stochastic matrices, and $\omega_{n}$ denote the convex polytope of all $n\times n$ doubly substochastic matrices. For a matrix $A\in\omega_n$, define the sub-defect of $A$ to be the smallest integer $k$ such that there exists an $(n+k)\times(n+k)$ doubly stochastic matrix containing $A$ as a submatrix. Let $\omega_{n,k}$ denote the subset of $\omega_n$ which contains all doubly substochastic matrices with sub-defect $k$. For $\pi$ a permutation of symmetric group of degree $n$, the sequence of elements $a_{1\pi(1)},a_{2\pi(2)}, \ldots, a_{n\pi(n)}$ is called the diagonal of $A$ corresponding to $\pi$. Let $h(A)$ and $l(A)$ denote the maximum and minimum diagonal sums of $A\in \omega_{n,k}$, respectively. In this paper, existing results of $h$ and $l$ functions are extended from $\Omega_n$ to $\omega_{n,k}.$ In addition, an analogue of Sylvesters law of the $h$ function on $\omega_{n,k}$ is proved.

scholarly journals Doubly stochastic matrices and the quantum channels

2021 ◽  
Vol 17 (1) ◽  
pp. 73-107
Author(s):  
H. K. Das ◽  
Md. Kaisar Ahmed
Keyword(s):  
Matrix Theory ◽  
Stochastic Matrix ◽  
Quantum Channels ◽  
Stochastic Matrices ◽  
Famous Conjecture ◽  
Frobenius Theorem ◽  
Doubly Stochastic Matrix ◽  
Birkhoff Theorem ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices

Abstract The main object of this paper is to study doubly stochastic matrices with majorization and the Birkhoff theorem. The Perron-Frobenius theorem on eigenvalues is generalized for doubly stochastic matrices. The region of all possible eigenvalues of n-by-n doubly stochastic matrix is the union of regular (n – 1) polygons into the complex plane. This statement is ensured by a famous conjecture known as the Perfect-Mirsky conjecture which is true for n = 1, 2, 3, 4 and untrue for n = 5. We show the extremal eigenvalues of the Perfect-Mirsky regions graphically for n = 1, 2, 3, 4 and identify corresponding doubly stochastic matrices. Bearing in mind the counterexample of Rivard-Mashreghi given in 2007, we introduce a more general counterexample to the conjecture for n = 5. Later, we discuss different types of positive maps relevant to Quantum Channels (QCs) and finally introduce a theorem to determine whether a QCs gives rise to a doubly stochastic matrix or not. This evidence is straightforward and uses the basic tools of matrix theory and functional analysis.

Nonnegative generalized doubly stochastic matrices with prescribed elementary divisors

2015 ◽  
Vol 30 ◽  
pp. 704-720
Author(s):  
Ricardo Soto ◽  
Elvis Valero ◽  
Mario Salas ◽  
Hans Nina
Keyword(s):  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Stochastic Matrices ◽  
Jordan Canonical Form ◽  
Doubly Stochastic ◽  
Elementary Divisors ◽  
Doubly Stochastic Matrices ◽  
Solution Matrix ◽  
Generalized Doubly Stochastic Matrix ◽  
Perron Eigenvalue

This paper provides sufficient conditions for the existence of nonnegative generalized doubly stochastic matrices with prescribed elementary divisors. These results improve previous results and the constructive nature of their proofs allows for the computation of a solution matrix. In particular, this paper shows how to transform a generalized stochastic matrix into a nonnegative generalized doubly stochastic matrix, at the expense of increasing the Perron eigenvalue, but keeping other elementary divisors unchanged. Under certain restrictions, nonnegative generalized doubly stochastic matrices can be constructed, with spectrum \Lambda = {\lambda_1,\lambda_2 2, . . . , \lambda_n} for each Jordan canonical form associated with \Lambda

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