An inverse problem for symmetric doubly stochastic matrices

Linear and Multilinear Algebra

2013

Self Cite

Section: Applications To Doubly Stochastic Matricesmentioning

confidence: 99%

Generalization of some results concerning eigenvalues of a certain class of matrices and some applications

Linear and Multilinear Algebra

2013

Self Cite

Linear Algebra and its Applications

“…Recently, the problem has gained new impetus as reflected by a plethora of new sufficient conditions ( [2,5,6,7,10,11]). We refer to Mourad's paper [9] for a good overview concerning the said problems.…”

Section: Introductionmentioning

confidence: 99%

Inverse problems for symmetric doubly stochastic matrices whose Suleĭmanova spectra are bounded below by 1/2

Gnacik

Kania

2020

A new sufficient condition for a list of real numbers to be the spectrum of a symmetric doubly stochastic matrix is presented; this is a contribution to the classical spectral inverse problem for symmetric doubly stochastic matrices that is still open in its full generality. It is proved that whenever λ 2 , . . . , λ n are non-positive real numbers with 1 + λ 2 + . . . + λ n 1/2, then there exists a symmetric, doubly stochastic matrix whose spectrum is precisely (1, λ 2 , . . . , λ n ). We point out that this criterion is incomparable to the classical sufficient conditions due to Perfect-Mirsky, Soules, and their modern refinements due to Nadar et al. We also provide some examples and applications of our results.Date: October 22, 2019. 2010 Mathematics Subject Classification. 65F18 (primary), and 15A18, 15A12 (secondary).

“…Of special importance are the nonnegative elements in Ω 1 (n, R) which are called the doubly stochastic matrices. The theory of doubly stochastic matrices is particularly endowed with a large collection of applications in other area of mathematics and also in other disciplines (see for example [1,2,3,6,7,19,25,28]). Let the set of all n × n doubly stochastic matrices be denoted by ∆ n and the set of all n × n symmetric elements in ∆ n will be denoted by ∆ s n .…”

Section: Introductionmentioning

confidence: 99%

“…Although doubly stochastic matrices have been studied extensively, the (DIEP) and (RDIEP) have been considered in [22,26] where all the results obtained are partial. The (SDIEP) was studied in [12,19,20,21,24], and earlier work can be found in [13,22,27] and all the results obtained are also partial. For general n, all three problems remain open.…”

Section: Introductionmentioning

confidence: 99%

An algorithm for constructing doubly stochastic matrices for the inverse eigenvalue problem

Abbas

Linear Algebra and its Applications

et al. 2013

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In this note, we present an algorithm that yields many new methods for constructing doubly stochastic and symmetric doubly stochastic matrices for the inverse eigenvalue problem. In addition, we introduce new open problems in this area that lay the ground for future work.(possibly complex) to be the spectrum of an n × n nonnegative matrix A. Although this inverse eigenvalue problem has attracted a considerable amount of interest, for n > 3 it is still unsolved except in restricted cases. Generally, we have two cases that result in three problems.• When λ = (λ 1 , ..., λ n ) is complex, little is known. The case n ≤ 3 was completely solved in [17] and the solution of the 4 × 4 trace-zero nonnegative inverse eigenvalue problem was given in [23].• When the n-tuples (λ 1 , . . . , λ n ) are real, then we have the following two problems: 1) The real nonnegative inverse eigenvalue problem (RNIEP) asks which sets of n real numbers occur as the spectrum of an n × n nonnegative matrix A.2) The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks which sets of n real numbers can occur as the spectrum of an n × n symmetric nonnegative matrix A.Each problem remains open. Many partial results for the three problems are known see the recent book [10] and the references therein, for the collection of all known results concerning these problems. Various people raised the question whether the (RNIEP) and (SNIEP) are generally equivalent. In the low dimension n ≤ 4, the two problems are actually equivalent (see [9]). However, for n > 4, the paper [14] showed that the two problems are generally different. In addition, the paper [9] gives a construction for data λ = (λ 1 , ..., λ 5 ) which is a solution for the (RNIEP) and there is no symmetric nonnegative 5 × 5 matrix with spectrum λ.Another object of study in this area that has a big interest is the inverse eigenvalue problem for nonnegative matrices with extra properties. For example, we can consider the same problems for doubly stochastic matrices. So that we have the following problems.Problem 1.1. The doubly stochastic inverse eigenvalue problem denoted by (DIEP), is the problem of determining the necessary and sufficient conditions for a complex n-tuples to be the spectrum of an n × n doubly stochastic matrix. Equivalently, this problem can also be characterized as the problem of finding the region Θ n of C n such that any point in Θ n is the spectrum of an n × n doubly stochastic matrix. Now, when the n-tuples (λ 1 , . . . , λ n ) are all real, then we have the following two problems: