Increasable doubly substochastic matrices with application to infinite linear equations

Eshkaftaki, Ali Bayati

doi:10.1080/03081087.2021.1939253

Cited by 2 publications

(3 citation statements)

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“…Particulary, if I is a finite set with n elements, we denote by Ω n and ω n for the set of all n × n doubly stochastic and doubly substochastic matrices, respectively. Definition 1.2 [3,11] Let A = [a ij ] be an I × I doubly substochastic matrix. Then A is called increasable if there exists an I × I doubly stochastic matrix D such that A ≤ D.…”

Section: Introductionmentioning

confidence: 99%

“…Definition 1.5 [3] Two functions f, g : I → [0, ∞), are said to be conjugate if there is a non-negative matrix A such that f = r A , and g = c A . We use the notation f ≈ g whenever f, g are conjugate.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.6 [3] Suppose that f, g : I → [0, ∞). Then f ≈ g if and only if one of the following conditions hold.…”

Section: Introductionmentioning

confidence: 99%

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A generalization of a theorem of von Neumann

Eshkaftaki

2023

Preprint

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In 1953 von Neumann has proved that every n × n doubly substochastic matrix A can be increased to a doubly stochastic matrix, i.e., there is an n × n doubly stochastic matrix D for which A ≤ D. In this paper, we will discuss this result for a class of I × I doubly substochastic matrices. In fact, by a constructive method, we find an equivalent condition for existence of a doubly stochastic matrix D which satisfies A ≤ D, for all A ∈ A, where A is assumed a class of (finite or infinite) doubly substochastic matrices. Such a matrix D is called a cover of A. The uniqueness of the cover will also be discussed. Then we obtain an application of this concept to a system of (infinite) linear equations and inequalities. MSC(2020): 15B48, 15B51, 15A45, 47A50

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%