The Spectra of Nonnegative Matrices Via Symbolic Dynamics

Boyle, Mike; Handelman, David

doi:10.2307/2944339

Cited by 143 publications

(112 citation statements)

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“…To find the solvability of these three problems, people have been studying them in the recent 70 years (refer to the references), the achievements people have got and their limitations and practical application depict were evaluated in scholarly treatise [7] (Moody T. Chu, Gene H. Golub, Inverse Eigenvalue Problems: Theory, Algorithms, and Applications, Oxford University press. P 93-122) and article [8] (Ricardo L.SoTo, Reliability by symmetric nonnegative matrces, http://www.scielo.cl/pdf/proy/v24n1/art06. pdf).…”

Section: Problem 3 (Sniep)mentioning

confidence: 99%

“…The first systematic treatment of eigenvalues of symmetric nonnegative matrices can probably be attributed to Fiedler (1974). A more comprehensive study was conducted by Boyle and Handelman (1991) using the notion of symbolic dynamics to characterize the conditions under which a given set is a portion of the spectrum of a nonnegative matrix or primitive matrix. …”

Section: Problem 3 (Sniep)mentioning

confidence: 99%

“…A short list of references giving various necessary or sufficient conditions includesBarrett and Johnson (1984), Boyle and Handelman (1991), Friedland (1978), Friedland and Melkman (1979), Loewy and London (1978), de Oliveira (1983), and Reams (1996). The difficulty is that the necessary condition is usually too general and the sufficient condition too specific.…”

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confidence: 99%

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On Open Problems of Nonnegative Inverse Eigenvalues Problem

Wu¹

2011

APM

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Section: Problem 3 (Sniep)mentioning

confidence: 99%

Section: Problem 3 (Sniep)mentioning

confidence: 99%

mentioning

confidence: 99%

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On Open Problems of Nonnegative Inverse Eigenvalues Problem

Wu¹

2011

APM

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“…Inverse eigenvalue problems are well-studied problems in linear algebra (see [5]) and perhaps the most important unsolved problem of this type is the nonnegative inverse eigenvalue problem, which asks when a set of numbers is the spectrum of a nonnegative matrix. Although the problem has been studied extensively, in general it remains unsolved (see, for example, [4]). …”

Section: The General Spectrummentioning

confidence: 99%

The quasimonotonicity of linear differential systems — the complex spectrum

Beaver

Canright

2001

Applicable Analysis

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The method of vector Lyapunov functions to determine stability in dynamical systems requires that the comparison system be quasimonotone nondecreasing with respect to a cone contained in the nonnegative orthant. For linear comparison systems in R n with real spectra, Heikkilä solved the problem for n = 2 and gave necessary conditions for n > 2. We previously showed a sufficient condition for n > 2, and here, for systems with complex eigenvalues, we give conditions for which the problem reduces to the nonnegative inverse eigenvalue problem.

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“…457-458], and Fiedler [4] and that if an equality occurs for some x > 1, then χ A (x) = x n+1 −1 as polynomials. In their celebrated paper [3], Boyle and Handelman make the following conjecture: Suppose further that A has a nonzero spectrum of size r + 1 ≥ 1. Then, (1.2) χ A (x) ≤ x n+1 − x n−r , for all x ≥ 1.…”

Section: Introductionmentioning

confidence: 99%

Untitled

2009

Proc. Amer. Math. Soc.

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Abstract. In their celebrated 1991 paper on the inverse eigenvalue problem for nonnegative matrices, Boyle and Handelman conjectured that if A is an (n+1)×(n+1) nonnegative matrix whose nonzero eigenvalues are:To date the status of this conjecture is that Ambikkumar and Drury (1997) showed that the conjecture is true when 2(r + 1) ≥ (n + 1), while Koltracht, Neumann, and Xiao (1993) showed that the conjecture is true when n ≤ 4 and when the spectrum of A is real. They also showed that the conjecture is asymptotically true with the dimension.Here we prove a slightly stronger inequality than in ( * ), from which it follows that the Boyle-Handelman conjecture is true. Actually, we do not start from the assumption that the λ i 's are eigenvalues of a nonnegative matrix, but that λ 1 , . . . , λ r+1 satisfy λ 0 ≥ |λ i |, i = 1, . . . , r, and the trace conditions:A strong form of the Boyle-Handelman conjecture, conjectured in 2002 by the present authors, says that ( * ) continues to hold if the trace inequalities in ( * * ) hold only for k = 1, . . . , r. We further improve here on earlier results of the authors concerning this stronger form of the Boyle-Handelman conjecture.

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The Spectra of Nonnegative Matrices Via Symbolic Dynamics

Cited by 143 publications

References 13 publications

On Open Problems of Nonnegative Inverse Eigenvalues Problem

On Open Problems of Nonnegative Inverse Eigenvalues Problem

The quasimonotonicity of linear differential systems — the complex spectrum

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