Allow problems concerning spectral properties of sign pattern matrices: A survey

Catral, Minerva; Olesky, D.D.; Driessche, P. van den

doi:10.1016/j.laa.2009.01.031

Cited by 34 publications

(22 citation statements)

References 49 publications

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“…Currently, the nilpotent-Jacobian method (see [2,4,5,11,17]) is often used to prove that a pattern is spectrally (and hence inertially) arbitrary. For a survey discussing sign patterns that are SAP, IAP, PN or PS, see [3]. More recently, constructions of {0, +, −}-patterns that are PS are given in [13] and constructions of {0, * }-patterns that are PN are given in [1].…”

Section: Cavers and S Fallatmentioning

confidence: 99%

“…In the literature, for example, see [3], properties that matrix patterns may allow or require have been defined in the following manner: a pattern A allows (resp., requires) a property P if some (resp., every) matrix A ∈ Q(A) has property P. In this paper, we consider the problems of a pattern being IAP (resp., rIAP or SAP) to be properties a pattern may allow in the sense that the pattern must allow all possible inertias (resp., refined inertias or spectra closed under complex conjugation). Note that these three spectral properties do not directly satisfy the above definition of allows.…”

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confidence: 99%

“…Sign patterns that are PN or PS have been well studied in the literature (see, for example, the survey paper [3]). Since sign patterns of order 2 that are PPI (resp., PN and PS) are known or easy to classify, by Lemma 2.3, taking relaxations then generates all such S-patterns of order 2 that are PPI (resp., PN and PS).…”

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confidence: 99%

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Allow problems concerning spectral properties of patterns

Cavers¹,

Fallat²

2012

ELA

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Abstract. Let S ⊆ {0, +, −, + 0 , − 0 , * , #} be a set of symbols, where + (resp., −, + 0 and − 0 ) denotes a positive (resp., negative, nonnegative and nonpositive) real number, and * (resp., #) denotes a nonzero (resp., arbitrary) real number. An S-pattern is a matrix with entries in S. In particular, a {0, +, −}-pattern is a sign pattern and a {0, * }-pattern is a zero-nonzero pattern. This paper extends the following problems concerning spectral properties of sign patterns and zero-nonzero patterns to S-patterns: spectrally arbitrary patterns; inertially arbitrary patterns; refined inertially arbitrary patterns; potentially nilpotent patterns; potentially stable patterns; and potentially purely imaginary patterns. Relationships between these classes of S-patterns are given and techniques that appear in the literature are extended. Some interesting examples and properties of patterns when # belongs to the symbol set are highlighted. For example, it is shown that there is a {0, +, #}-pattern of order n that is spectrally arbitrary with exactly 2n − 1 nonzero entries. Finally, a modified version of the nilpotent-Jacobian method is presented that can be used to show a pattern is inertially arbitrary.

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Section: Cavers and S Fallatmentioning

confidence: 99%

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confidence: 99%

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Allow problems concerning spectral properties of patterns

Cavers¹,

Fallat²

2012

ELA

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“…Identifying potentially nilpotent patterns is partially motivated by the study of spectrally arbitrary patterns (see for example [1,3,11,13]). An n × n pattern A is said to be a spectrally arbitrary pattern over F if for every degree n monic polynomial q(x) with coefficients in F, there is a realization of A whose characteristic polynomial equals q(x).…”

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confidence: 99%

“…Van Tuyl was also identified in [7]. The paper [3] summarizes much of the current knowledge about potentially nilpotent patterns over R, while the more recent paper [1] introduces some new constructions for patterns associated with trees.…”

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confidence: 99%

Structure of nilpotent matrices over fields

Campbell¹,

Meulen²,

Tuyl³

2011

ELA

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Abstract. A zero-nonzero pattern A is said to be potentially nilpotent over a field F if there exists a nilpotent matrix with entries in F having zero-nonzero pattern A. We explore the construction of potentially nilpotent patterns over a field. We present classes of patterns which are potentially nilpotent over a field F if and only if the field F contains certain roots of unity. We then introduce some sparse patterns of order n ≥ 4 which are spectrally arbitrary over C but not over R. We also identify all irreducible patterns of order four which are potentially nilpotent over R or C.

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On Matrix Stability and Ecological Models

McGrane-Corrigan,

Mason

2023

Entomology in Focus

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Allow problems concerning spectral properties of sign pattern matrices: A survey

Cited by 34 publications

References 49 publications

Allow problems concerning spectral properties of patterns

Allow problems concerning spectral properties of patterns

Structure of nilpotent matrices over fields

On Matrix Stability and Ecological Models

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