Generating potentially nilpotent full sign patterns

Ij, Kim; Olesky, D.D.; Shader, Bryan L.; Driessche, Van den; Holst, van der H Hein; Meulen, KN Vander

doi:10.13001/1081-3810.1302

Cited by 21 publications

(20 citation statements)

References 17 publications

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“…In [9] the question was raised as to whether or not a sign pattern that is an rIAP must be a SAP. In Section 2 we address this question for irreducible zero-nonzero patterns and exhibit such a pattern of smallest order that is an rIAP but not a SAP.…”

Section: Lemma 12 ([8 Proposition 2])mentioning

confidence: 99%

Refined inertially and spectrally arbitrary zero-nonzero patterns

Deaett¹,

Olesky²,

Driessche³

2010

ELA

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Abstract. The refined inertia of a matrix is a quadruple specifying its inertia and additionally the number of its eigenvalues equal to zero. Spectral properties, especially the refined inertias, of real matrices with a given zero-nonzero pattern are investigated. It is shown that every zero-nonzero refined inertially arbitrary pattern of order 4 or less is also spectrally arbitrary. Irreducible and reducible examples are presented to show that for higher orders this is not the case. A further example shows that two zero-nonzero patterns that are not refined inertially arbitrary can have a direct sum that is refined inertially arbitrary, paralleling a known result for inertially arbitrary patterns. Analogously, it is shown that the direct sum of two zero-nonzero patterns may be spectrally arbitrary even if neither summand is spectrally arbitrary.

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Section: Lemma 12 ([8 Proposition 2])mentioning

confidence: 99%

Refined inertially and spectrally arbitrary zero-nonzero patterns

Deaett¹,

Olesky²,

Driessche³

2010

ELA

Self Cite

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“…Similarly, a pattern A is said to be PPI + if A is potentially purely imaginary and G(A) allows a positive loop, allows a negative loop, and allows a negative 2-cycle. The property PN + was defined in [14] for sign patterns but explicitly required that G(A) has at least two loops. In our definition, it is possible that A can be PN + (or PPI + ) with the property that G(A) has exactly one loop with sign #.…”

Section: Cavers and S Fallatmentioning

confidence: 99%

“…In our definition, it is possible that A can be PN + (or PPI + ) with the property that G(A) has exactly one loop with sign #. The definition of PN + from [14] is motivated in part by Theorem 2.4 below, which is a generalization of [5,Lemma 5.1]. The definition of PN + in this paper is motivated by Corollary 2.8 and is equivalent to the definition in [14] for {0, +, −, * }-patterns.…”

Section: Cavers and S Fallatmentioning

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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“…As defined in [7], the refined inertia ri(A) of a real n × n matrix A is the ordered 4-tuple (n + , n − , n z , 2n p ) such that n + (resp., n − ) is the number of eigenvalues (including multiplicities) of A with positive (resp., negative) real part, and n z (resp., 2n p ) is the number of zero eigenvalues (resp., nonzero pure imaginary eigenvalues) of A. Here n + + n − + n z + 2n p = n. The inertia of A is (n + , n − , n z + 2n p ), thus the refined inertia subdivides those eigenvalues with zero real part and distinguishes between those that are exactly zero and those that are nonzero.…”

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confidence: 99%

Refined inertias of tree sign-patterns

Garnett¹,

Olesky²,

Driessche³

2013

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Abstract. The refined inertia (n + , n − , nz, 2np) of a real matrix is the ordered 4-tuple that subdivides the number n 0 of eigenvalues with zero real part in the inertia (n + , n − , n 0 ) into those that are exactly zero (nz) and those that are nonzero (2np). For n ≥ 2, the set of refined inertias Hn = {(0, n, 0, 0), (0, n − 2, 0, 2), (2, n − 2, 0, 0)} is important for the onset of Hopf bifurcation in dynamical systems. Tree sign patterns of order n that require or allow the refined inertias Hn are considered. For n = 4, necessary and sufficient conditions are proved for a tree sign pattern (necessarily a path or a star) to require H 4 . For n ≥ 3, a family of n × n star sign patterns that allows Hn is given, and it is proved that if a star sign pattern requires Hn, then it must have exactly one zero diagonal entry associated with a leaf in its digraph.

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Generating potentially nilpotent full sign patterns

Cited by 21 publications

References 17 publications

Refined inertially and spectrally arbitrary zero-nonzero patterns

Refined inertially and spectrally arbitrary zero-nonzero patterns

Allow problems concerning spectral properties of patterns

Refined inertias of tree sign-patterns

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