A unified view on compensation criteria in the real nonnegative inverse eigenvalue problem

A list Λ = {λ 1 , λ 2 , . . . , λ n } of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. The list Λ is said to be universally realizable (U R) if it is the spectrum of a nonnegative matrix for each possible Jordan canonical form allowed by Λ. It is well known that an n × n nonnegative matrix A is co-spectral to a nonnegative matrix B with constant row sums. In this paper, we extend the co-spectrality between A and B to a similarity between A and B, when the Perron eigenvalue is simple. We also show that if ǫ ≥ 0 and Λ = {λ 1 , λ 2 , . . . , λ n } is U R, then {λ 1 + ǫ, λ 2 , . . . , λ n } is also U R. We give counter-examples for the cases: Λ = {λ 1 , λ 2 , . . . , λ n } is U R implies {λ 1 + ǫ, λ 2 − ǫ, λ 3 , . . . , λ n } is U R, and Λ 1 , Λ 2 are U R implies Λ 1 ∪ Λ 2 is U R.

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“…which has a positive eigenvector [1,1,2] associated to the double eigenvalue λ 1 = 2. Now, suppose that A is a block diagonal matrix.…”

Section: Nonnegative Matrices Similar To Nonnegative Matrices With Comentioning

confidence: 99%

On universal realizability of spectra

Julio

Marijuán

Pisonero

et al. 2019

Linear Algebra and its Applications

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“…Other well-known sufficient conditions manipulate certain spectra to get a new realizable spectrum: Guo in 1997 [40, Theorems 2.1 and 3.1] or C-realizability [11] that we name the game condition [85].…”

Section: Sufficient Conditionsmentioning

confidence: 99%

The NIEP

Johnson

Marijuán

Paparella

et al. 2018

Operator Theory, Operator Algebras, and Matrix Theory

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The nonnegative inverse eigenvalue problem (NIEP) asks which lists of n complex numbers (counting multiplicity) occur as the eigenvalues of some n-by-n entry-wise nonnegative matrix. The NIEP has a long history and is a known hard (perhaps the hardest in matrix analysis?) and sought after problem. Thus, there are many subproblems and relevant results in a variety of directions. We survey most work on the problem and its several variants, with an emphasis on recent results, and include 130 references. The survey is divided into: a) the single eigenvalue problems; b) necessary conditions; c) low dimensional results; d) sufficient conditions; e) appending 0's to achieve realizability; f) the graph NIEP's; g) Perron similarities; and h) the relevance of Jordan structure. Mathematics Subject Classification (2010). Primary 15A18, 15A29, 15A42.

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“…In [6], Borobia, Moro and Soto construct realisable lists in the RNIEP, starting from trivially realisable lists, using three well-known results.…”

Section: C-realisability and The Rniepmentioning

confidence: 99%

“…We also consider a sufficient condition for the RNIEP due to Borobia, Moro and Soto [6] called "C-realisability" and a family of sufficient conditions for the SNIEP due to Soto [7]. We show that C-realisability is also sufficient for the SNIEP and that σ is C-realisable if and only if it satisfies one of Soto's conditions.…”

Section: Introductionmentioning

confidence: 96%

Connecting sufficient conditions for the Symmetric Nonnegative Inverse Eigenvalue Problem

Ellard

Šmigoc

2016

Linear Algebra and its Applications

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We say that a list of real numbers is "symmetrically realisable" if it is the spectrum of some (entrywise) nonnegative symmetric matrix. The Symmetric Nonnegative Inverse Eigenvalue Problem (SNIEP) is the problem of characterising all symmetrically realisable lists.In this paper, we present a recursive method for constructing symmetrically realisable lists. The properties of the realisable family we obtain allow us to make several novel connections between a number of sufficient conditions developed over forty years, starting with the work of Fiedler in 1974. We show that essentially all previously known sufficient conditions are either contained in or equivalent to the family we are introducing.

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A unified view on compensation criteria in the real nonnegative inverse eigenvalue problem

Cited by 19 publications

References 8 publications

On universal realizability of spectra

On universal realizability of spectra

The NIEP

Connecting sufficient conditions for the Symmetric Nonnegative Inverse Eigenvalue Problem

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