Eigenvalues of nonnegative symmetric matrices

Fiedler, Miroslav

doi:10.1016/0024-3795(74)90031-7

Cited by 146 publications

(143 citation statements)

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“…Using the procedure in Remark 3.3, construct a 3 × 3 symmetric nonnegative matrix T with eigenvalues µ, λ 3 , λ 4 and diagonal entries ω 2 , ω 3 , ω 4 . Notice that these eigenvalues and diagonal entries satisfy the necessary and sufficient conditions given in Lemma 3.2:…”

Section: Construct the 2 × 2 Symmetric Nonnegative Matrixmentioning

confidence: 99%

“…Kellogg's criterion [8], for instance, was shown by Fiedler [4] to imply symmetric realizability. Radwan [16] proved that Borobia's criterion [1] is also a criterion for symmetric realizability , and Soto's criterion for the RNIEP [19] (Theorem 4.2 below), which contains both Kellogg's and Borobia's criteria [20], was shown in [23] to be also a symmetric realizability criterion.…”

Section: Comparison With Previous Criteriamentioning

confidence: 99%

“…Both problems, RNIEP and SNIEP, are equivalent for n ≤ 4 (see [26]), but are different otherwise [7]. Partial results for the SNIEP have been obtained in [4,24,16,21,23] (see [3, §2.2] and references therein for more on the SNIEP).…”

Section: Introductionmentioning

confidence: 99%

“…The case t = 3. There are also necessary and sufficient conditions, obtained by Fiedler [4], for the existence of a 3 × 3 symmetric nonnegative matrix with prescribed spectrum and diagonal entries: …”

mentioning

confidence: 99%

“…The matrix B for t = 3. Using the proof of Theorem 4.4 in [4], one can write a procedure to construct the symmetric nonnegative matrix B in Lemma 3.2. The procedure is as follows:…”

mentioning

confidence: 99%

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Symmetric nonnegative realization of spectra

Soto¹,

Rojo²,

Moro³

et al. 2007

ELA

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Abstract.A perturbation result, due to R. Rado and presented by H. Perfect in 1955, shows how to modify r eigenvalues of a matrix of order n, r ≤ n, via a perturbation of rank r, without changing any of the n − r remaining eigenvalues. This result extended a previous one, due to Brauer, on perturbations of rank r = 1. Both results have been exploited in connection with the nonnegative inverse eigenvalue problem. In this paper a symmetric version of Rado's extension is given, which allows us to obtain a new, more general, sufficient condition for the existence of symmetric nonnegative matrices with prescribed spectrum.Key words. Symmetric nonnegative inverse eigenvalue problem.

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Section: Construct the 2 × 2 Symmetric Nonnegative Matrixmentioning

confidence: 99%

Section: Comparison With Previous Criteriamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

“…The matrix B for t = 3. Using the proof of Theorem 4.4 in [4], one can write a procedure to construct the symmetric nonnegative matrix B in Lemma 3.2. The procedure is as follows:…”

mentioning

confidence: 99%

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Symmetric nonnegative realization of spectra

Soto¹,

Rojo²,

Moro³

et al. 2007

ELA

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A family of realizability criteria for the real and symmetric nonnegative inverse eigenvalue problem

Soto

2011

Numerical Linear Algebra App

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A new realizability criterion for the real nonnegative inverse eigenvalue problem is introduced. This criterion is a nontrivial extension of a powerful previous sufficient condition, based on negativity compensation. If the criterion is satisfied, then we can always construct a realizing matrix. It is also proved that this new criterion is easily adaptable to be sufficient for the construction of a symmetric nonnegative matrix with given spectrum. In a natural way, the criterion extends to a family of sufficient conditions for the problem to have a solution.

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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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Eigenvalues of nonnegative symmetric matrices

Cited by 146 publications

References 5 publications

Symmetric nonnegative realization of spectra

Symmetric nonnegative realization of spectra

A family of realizability criteria for the real and symmetric nonnegative inverse eigenvalue problem

The NIEP

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