On the symmetric doubly stochastic inverse eigenvalue problem

Lei, Yingjie; Xu, Wei-Ru; Lu, Yilong; Niu, Yan-Ru; Gu, Xian-Ming

doi:10.1016/j.laa.2013.12.005

Cited by 19 publications

(10 citation statements)

References 27 publications

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“…There has been a lot of interest in this kind of doubly stochastic inverse eigenvalue problem. The reader may refer to Zhao et al (2016), Chu and Golub (2005), Yao et al (2016), Adelia et al (2018), Lei et al (2014), Hwang and Pyo (2004) for various results. In this paper, we consider another kind of doubly stochastic inverse eigenvalue problem (PDSIEP) based on partial information on eigenvalues and eigenvectors, which can be stated as follows.…”

Section: Introductionmentioning

confidence: 99%

Alternating projection method for doubly stochastic inverse eigenvalue problems with partial eigendata

Chen

Weng

2021

Comp. Appl. Math.

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In this paper, we consider doubly stochastic inverse eigenvalue problems with partial eigendata, which aims to construct a doubly stochastic matrix from the prescribed partial eigendata. We propose the alternating projection method for two closed convex sets to solve the doubly stochastic inverse eigenvalue problems. And each subproblem in the alternating projection method can be solved easily. Under the assumption that the intersection of the two closed convex sets is not empty, the convergence of the alternating projection method is proved. Numerical results illustrate the effectiveness of our method.

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Section: Introductionmentioning

confidence: 99%

Alternating projection method for doubly stochastic inverse eigenvalue problems with partial eigendata

Chen

Weng

2021

Comp. Appl. Math.

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“…The inverse eigenvalue problems for non-negative doubly stochastic matrices have its origin in work of [2][3][4][5]. For more details on inverse eigenvalue problems, we refer the reader to [6][7][8][9][10] and references therein.…”

Section: Introductionmentioning

confidence: 99%

On Spectral Properties of Doubly Stochastic Matrices

et al. 2020

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The relationship among eigenvalues, singular values, and quadratic forms associated with linear transforms of doubly stochastic matrices has remained an important topic since 1949. The main objective of this article is to present some useful theorems, concerning the spectral properties of doubly stochastic matrices. The computation of the bounds of structured singular values for a family of doubly stochastic matrices is presented by using low-rank ordinary differential equations-based techniques. The numerical computations illustrating the behavior of the method and the spectrum of doubly stochastic matrices is then numerically analyzed.

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“…The symmetric nonnegative inverse eigenvalue problem is one specific problem in a wide variety of inverse problems in linear algebra. The surveys [9], [17] and more recently [26], [32], [29] give a broad overview of the different inverse eigenvalue subproblems, their applications and the existing strategies to address them. Closely related to the SNIEP are the nonnegative inverse eigenvalue problem (NIEP) for nonnegative matrices and the symmetric doubly stochastic inverse eigenvalue problem (SDIEP) for symmetric nonnegative matrices with row and column sums equal to one.…”

Section: Introductionmentioning

confidence: 99%

“…This strategy, together with Soules' approach, leads to the most general sufficient conditions for the SNIEP [18], [32], [26]. For a more complete overview of the SNIEP and other inverse eigenvalue problems, we again refer to the excellent surveys [9], [17], [26], [32], [29].…”

Section: Introductionmentioning

confidence: 99%

Constructing Laplacian matrices with Soules vectors: inverse eigenvalue problem and applications

Devriendt,

Lambiotte,

Van Mieghem

2019

Preprint

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The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks which sets of numbers (counting multiplicities) can be the eigenvalues of a symmetric matrix with nonnegative entries. While examples of such matrices are abundant in linear algebra and various applications, this question is still open for matrices of dimension N ≥ 5. One of the approaches to solve the SNIEP was proposed by George W. Soules [45], relying on a specific type of eigenvectors (Soules vectors) to derive sufficient conditions for this problem. Elsner et al. [19] later showed a canonical way to construct all Soules vectors, based on binary rooted trees. While Soules vectors are typically treated as a totally ordered set of vectors, we propose in this article to consider a relaxed alternative: a partially ordered set of Soules vectors. We show that this perspective enables a more complete characterization of the sufficient conditions for the SNIEP. In particular, we show that the set of eigenvalues that satisfy these sufficient conditions is a convex cone, with symmetries corresponding to the automorphisms of the binary rooted tree from which the Soules vectors were constructed. As a second application, we show how Soules vectors can be used to construct graph Laplacian matrices with a given spectrum and describe a number of interesting connections with the concepts of hierarchical random graphs, equitable partitions and effective resistance.

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On the symmetric doubly stochastic inverse eigenvalue problem

Cited by 19 publications

References 27 publications

Alternating projection method for doubly stochastic inverse eigenvalue problems with partial eigendata

Alternating projection method for doubly stochastic inverse eigenvalue problems with partial eigendata

On Spectral Properties of Doubly Stochastic Matrices

Constructing Laplacian matrices with Soules vectors: inverse eigenvalue problem and applications

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