A note on the real inverse spectral problem for doubly stochastic matrices

The real (resp. symmetric) doubly stochastic inverse spectral problem is the problem of determining necessary and sufficient conditions for a real n-tuple λ=(1,λ2,...,λn) to be the spectrum of an n×n (resp. symmetric) doubly stochastic matrix. If λi≤0 for all i=2,...,n and the sum of all the entries...

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Bibliographic Details
Published in:Linear algebra and its applications 2019-05, Vol.569, p.206-240
Main Authors: Nader, Rafic, Mourad, Bassam, Bretto, Alain, Abbas, Hassan
Format: Article
Language:English
Subjects:
Computer Science
Economic models
Inverse eigenvalue problem
Linear algebra
Mathematical analysis
Matrix methods
Normalized Suleimanova spectrum
Realizability
Spectra
Symmetric doubly stochastic matrices
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Summary:The real (resp. symmetric) doubly stochastic inverse spectral problem is the problem of determining necessary and sufficient conditions for a real n-tuple λ=(1,λ2,...,λn) to be the spectrum of an n×n (resp. symmetric) doubly stochastic matrix. If λi≤0 for all i=2,...,n and the sum of all the entries in λ is nonnegative, then we refer to such λ as a normalized Suleimanova spectrum. The purpose of this paper is to first fix an error in Theorem 9 of the recent paper Adeli et al. (2018) [1], after giving a counterexample. Secondly, we give a negative answer to a question posed in Johnson and Paparella (2016) [3] concerning the realizability of normalized Suleimanova spectra for the case when n is odd. Some sufficient conditions for a positive answer to this question are given.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2019.01.017