Perturbations on constructible lists preserving realizability in the NIEP and questions of Monov

Cronin, Anthony; Laffey, Thomas J.

doi:10.1016/j.laa.2013.12.003

Cited by 3 publications

(4 citation statements)

References 14 publications

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“…, λn} is a realizable list of real numbers, of the form (3) or (4), then Λ is permutatively realizable. In [3,18], the authors prove certain perturbation results, for spectra realizable by circulant matrices. In particular, from the results in [18], and for the spectrum , where Thus,…”

Section: Permutative Realizabilitymentioning

confidence: 94%

Permutative universal realizability

2021

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A list of complex numbers Λ is said to be realizable, if it is the spectrum of a nonnegative matrix. In this paper we provide a new sufficient condition for a given list Λ to be universally realizable (UR), that is, realizable for each possible Jordan canonical form allowed by Λ. Furthermore, the resulting matrix (that is explicity provided) is permutative, meaning that each of its rows is a permutation of the first row. In particular, we show that a real Suleĭmanova spectrum, that is, a list of real numbers having exactly one positive element, is UR by a permutative matrix.

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Section: Permutative Realizabilitymentioning

confidence: 94%

Permutative universal realizability

2021

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“…A proof of Proposition 5.11 is immediate once a unitary matrix is provided that satisfies property (8). Recall that an n-by-n matrix H is called a Hadamard matrix (of order n) if h ij ∈ {±1} and HH ⊤ = nI.…”

Section: Differentiators and Complex Hadamard Matricesmentioning

confidence: 99%

“…It is clear that Johnson's conjecture implies Monov's conjecture. Cronin and Laffey [8] established Johnson's conjecture in the cases when n ≤ 4, or when n ≤ 6 and s 1 (Λ) = 0, .…”

Section: Introductionmentioning

confidence: 99%

On the realizability of the critical points of a realizable list

Hoover

McCormick

Paparella

et al. 2018

Linear Algebra and its Applications

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The nonnegative inverse eigenvalue problem (NIEP) is to characterize the spectra of entrywise nonnegative matrices. A finite multiset of complex numbers is called realizable if it is the spectrum of an entrywise nonnegative matrix. Monov conjectured that the k th -moments of the list of critical points of a realizable list are nonnegative. Johnson further conjectured that the list of critical points must be realizable. In this work, Johnson's conjecture, and consequently Monov's conjecture, is established for a variety of important cases including Ciarlet spectra, Suleȋmanova spectra, spectra realizable via companion matrices, and spectra realizable via similarity by a complex Hadamard matrix. Additionally we prove a result on differentiators and trace vectors, and use it to provide an alternate proof of a result due to Malamud and a generalization of a result due to Kushel and Tyaglov on circulant matrices. Implications for further research are discussed.

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“…We would like to thank the referee for providing the references [3] and [7]. The research of Li and Poon was supported by USA NSF, and HK RGC.…”

Section: Acknowledgmentmentioning

confidence: 99%

Perturbing eigenvalues of nonnegative matrices

Wang

Poon

2016

Linear Algebra and its Applications

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Let A be an irreducible (entrywise) nonnegative nˆn matrix with eigenvalues ρ, λ 2 " b`ic, λ 3 " b´ic, λ 4 ,¨¨¨, λ n , where ρ is the Perron eigenvalue. It is shown that for any t P r0, 8q there is a nonnegative matrix with eigenvalues ρ`t, λ 2`t , λ 3`t , λ 4¨¨¨, λ n , whenevert ě γ n t with γ 3 " 1, γ 4 " 2, γ 5 " ? 5 and γ n " 2.25 for n ě 6. The result improves that of Guo et al. Our proof depends on an auxiliary result in geometry asserting that the area of an n-sided convex polygon is bounded by γ n times the maximum area of a triangle lying inside the polygon.

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Perturbations on constructible lists preserving realizability in the NIEP and questions of Monov

Cited by 3 publications

References 14 publications

Permutative universal realizability

Permutative universal realizability

On the realizability of the critical points of a realizable list

Perturbing eigenvalues of nonnegative matrices

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