An inequality for the spectra of nonnegative matrices

Abstract:In this article we provide some lists of real numbers which can be realized as the spectra of nonnegative diagonalizable matrices but which are not the spectra of nonnegative symmetric matrices. In particular, we examine the classical list σ = ( + t, − t, − , − , − ) with t ≥ , and show that σ is realizable by a nonnegative diagonalizable matrix only for t ≥ . We also provide examples of lists which are realizable as the spectra of nonnegative matrices, but not as the spectra of nonnegative diagonalizable matrices by examining the Jordan Normal Form.

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“…A necessary condition for NIEP for general n involving only the rst three Newton power sums s k is given by Cronin and La ey in [4].…”

Section: Necessary Conditionsmentioning

confidence: 99%

“…A new necessary condition for the SNIEP when the trace is at least half the spectral radius is given by Loewy and Spector in [18]. A necessary condition for general n involving only the first three power sums s k is given by Cronin and Laffey in [1].…”

Section: Necessary Conditionsmentioning

confidence: 99%

The Diagonalizable Nonnegative Inverse Eigenvalue Problem

Cronin

Laffey

2018

Special Matrices

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“…In the applications of this lemma, we will choose A, K and L in such a way that M has a structure which makes its characteristic polynomial easy to compute. Several such structured matrices-such as companion matrices, doubly companion matrices and block companion matrices-have been studied in the context of the NIEP, for example by Friedland, Laffey,Šmigoc and Cronin [2], [9], [1] and indeed, the form of the matrix C in Lemma 2.3 has been chosen with such matrices in mind.…”

Section: Now Definementioning

confidence: 99%

Constructing new realisable lists from old in the NIEP

Ellard

Šmigoc

2014

Linear Algebra and its Applications

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“…Nonetheless, the problem is very attractive and has been attacked by many excellent researchers (see references). There have been prior surveys of work on the NIEP ( [129,93,4,28,7,21,118]), but none recently. With a great deal of important recent activity, a new survey, emphasizing this activity, will be welcome to anyone considering inquiry in this area.…”

Section: Introductionmentioning

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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An inequality for the spectra of nonnegative matrices

Cited by 6 publications

References 7 publications

The Diagonalizable Nonnegative Inverse Eigenvalue Problem

The Diagonalizable Nonnegative Inverse Eigenvalue Problem

Constructing new realisable lists from old in the NIEP

The NIEP

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