Inertially arbitrary sign patterns with no nilpotent realization

Kim, In-Jae; Olesky, D.D.; Driessche, P. van den

doi:10.1016/j.laa.2006.07.010

Cited by 16 publications

(18 citation statements)

References 10 publications

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“…Except possibly for the statement "A is an rIAP implies that A is a SAP", the converse of each implication in the diagram above is, in general, false. A family of sign patterns of odd order n ≥ 5 that is inertially arbitrary but not potentially nilpotent is presented in [12]. …”

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confidence: 99%

Generating potentially nilpotent full sign patterns

Ij¹,

Olesky²,

Shader³

et al. 2009

ELA

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Abstract.A sign pattern is a matrix with entries in {+, −, 0}. A full sign pattern has no zero entries. The refined inertia of a matrix pattern is defined and techniques are developed for constructing potentially nilpotent full sign patterns. Such patterns are spectrally arbitrary. These techniques can also be used to construct potentially nilpotent sign patterns that are not full, as well as potentially stable sign patterns.

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mentioning

confidence: 99%

Generating potentially nilpotent full sign patterns

Ij¹,

Olesky²,

Shader³

et al. 2009

ELA

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“…Therefore, by Theorem 2.13, every superpattern ofĈ n is IAP. In fact,Ĉ n is a minimal IAP with respect to both signings and subpatterns ( [15,Lemma 20] implies any signing ofĈ n is not IAP).…”

Section: 3mentioning

confidence: 99%

“…Necessary conditions on the signs of principal minors for an IAP can be found in [7,Lemma 1] and [15,Lemma 20].…”

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confidence: 99%

Allow problems concerning spectral properties of patterns

Cavers¹,

Fallat²

2012

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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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“…al. [10,Theorem 1] in which a sufficient condition for a pattern to be inertially arbitrary is found with respect to the set of polynomials that a pattern realizes. We use the fact that if A is a matrix of order n with eigenvalues λ 1 , λ 2 , .…”

Section: Elamentioning

confidence: 99%

Inertially arbitrary nonzero patterns of order 4

Cavers¹,

Meulen²

2007

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Abstract. Inertially arbitrary nonzero patterns of order at most 4 are characterized. Some of these patterns are demonstrated to be inertially arbitrary but not spectrally arbitrary. The order 4 sign patterns which are inertially arbitrary and have a nonzero pattern that is not spectrally arbitrary are also described. There exists an irreducible nonzero pattern which is inertially arbitrary but has no signing that is inertially arbitrary. In fact, up to equivalence, this pattern is unique among the irreducible order 4 patterns with this property.

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Inertially arbitrary sign patterns with no nilpotent realization

Cited by 16 publications

References 10 publications

Generating potentially nilpotent full sign patterns

Generating potentially nilpotent full sign patterns

Allow problems concerning spectral properties of patterns

Inertially arbitrary nonzero patterns of order 4

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