Potentially eventually exponentially positive sign patterns

Archer, Marie; Catral, Minerva; Erickson, Craig; Haber, Rana; Hogben, Leslie; Martínez-Rivera, Xavier; Ochoa, Antonio

doi:10.2140/involve.2013.6.261

Cited by 3 publications

(12 citation statements)

References 4 publications

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“…It is known from [7] that sign pattern A is PEEP if the positive part A + is irreducible. Since every superpattern of PEEP sign patterns is also PEEP, it is necessary to investigate the minimality of PEEP sign patterns.…”

Section: Terminologies and Preliminary Resultsmentioning

confidence: 99%

“…It is known from [7] that the converse of Lemma 2 is not true. But for the sign patterns with only − on the diagonal, the converse is true.…”

Section: Propositionmentioning

confidence: 96%

“…Recall that a real eigenvalue γ ∈ σ(A) (the spectrum of A) is called rightmost, if γ is simple, and Re(λ) < γ for all λ ∈ σ(A) (λ = γ); see [7] for details. The following Lemma 1 follows readily from the Theorem 3.3 in [2] and Proposition 1.6 in [7] Lemma 1. Let A be an n × n real matrix.…”

Section: Propositionmentioning

confidence: 99%

“…Following [7], let A D(+) (respectively, A D(0) and A D(−) ) be the sign pattern whose off-diagonal entries are the same as A and all the diagonal entries are + (respectively, 0 and −). Observation 1.4 in [7] is cited here as Lemma 2 for the sake of convenience.…”

Section: Propositionmentioning

confidence: 99%

“…An n × n sign pattern A is called potentially eventually exponentially positive (PEEP), if there exists some A ∈ Q(A) such that A is EEP. Some sufficient conditions and some necessary conditions for an n × n sign pattern to be PEEP have been established in [7]. However, there are still not any characterization about the PEEP sign patterns.…”

Section: Introductionmentioning

confidence: 99%

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On the Eventual Exponential Positivity of Some Tree Sign Patterns

2021

Symmetry

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An n×n matrix A is called eventually exponentially positive (EEP) if etA=∑k=0∞tkAkk!>0 for all t≥t0, where t0≥0. A matrix whose entries belong to the set {+,−,0} is called a sign pattern. An n×n sign pattern A is called potentially eventually exponentially positive (PEEP) if there exists some real matrix realization A of A that is EEP. Characterizing the PEEP sign patterns is a longstanding open problem. In this article, A is called minimally potentially eventually exponentially positive (MPEEP), if A is PEEP and no proper subpattern of A is PEEP. Some preliminary results about MPEEP sign patterns and PEEP sign patterns are established. All MPEEP sign patterns of orders n≤3 are identified. For the n×n tridiagonal sign patterns Tn, we show that there exists exactly one MPEEP tridiagonal sign pattern Tno. Consequently, we classify all PEEP tridiagonal sign patterns as the superpatterns of Tno. We also classify all PEEP star sign patterns Sn and double star sign patterns DS(n,m) by identifying all the MPEEP star sign patterns and the MPEEP double star sign patterns, respectively.

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Section: Terminologies and Preliminary Resultsmentioning

confidence: 99%

“…It is known from [7] that the converse of Lemma 2 is not true. But for the sign patterns with only − on the diagonal, the converse is true.…”

Section: Propositionmentioning

confidence: 96%

Section: Propositionmentioning

confidence: 99%

Section: Propositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Eventual Exponential Positivity of Some Tree Sign Patterns

2021

Symmetry

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Sign patterns that require eventual exponential nonnegativity

Erickson

2015

ELA

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Abstract. Sign patterns that require exponential nonnegativity are characterized. A set of conditions necessary for a sign pattern to require eventual exponential nonnegativity are established. It is shown that these conditions are also sufficient for an upper triangular sign pattern to require eventual exponential nonnegativity and it is conjectured that these conditions are both necessary and sufficient for any sign pattern to require eventual exponential nonnegativity. It is also shown that the maximum number of negative entries in a sign pattern that requires eventual exponential nonnegativity is (n−1)(n−2) 2 + 2.

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Eventual Positivity of a Class of Double Star-like Sign Patterns

Mou

et al. 2022

Symmetry

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Identifying and classifying the potentially eventually positive sign patterns and the potentially eventually exponentially positive sign patterns of orders greater than 3 have been raised as two open problems since 2010. In this article, we investigate the potential eventual positivity of the class of double star-like sign patterns S(n,m,1) whose underlying graph G(S(n,m,1)) is obtained from the underlying graph G(S(n,m)) of the (n+m)-by-(n+m) double star sign patterns S(n,m) by adding an additional vertex adjacent to the two center vertices and removing the edge between the center vertices. We firstly establish some necessary conditions for a double star-like sign pattern to be potentially eventually positive, and then identify all the minimal potentially eventually positive double star-like sign patterns. Secondly, we classify all the potentially eventually positive sign patterns in the class of double star-like sign patterns S(n,m,1). Finally, as an application of our results about the potentially eventually positive double star-like sign patterns, we identify all the minimal potentially eventually exponentially positive sign patterns and characterize all the potentially eventually exponentially positive sign patterns in the class of double star-like sign patterns S(n,m,1).

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Potentially eventually exponentially positive sign patterns

Cited by 3 publications

References 4 publications

On the Eventual Exponential Positivity of Some Tree Sign Patterns

On the Eventual Exponential Positivity of Some Tree Sign Patterns

Sign patterns that require eventual exponential nonnegativity

Eventual Positivity of a Class of Double Star-like Sign Patterns

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