Star Sign Patterns Research Articles

Sign patterns associated with some graphs that allow or require diagonalizability

The problems of characterizing sign pattern matrices that allow or require diagonalizability are mostly open. In this paper, we introduce the concept of essential index for a tree sign pattern matrix and use it to investigate the allow problem on diagonalizability for sign pattern matrices having their graphs as trees. We characterize sign pattern matrices allowing diagonalizability, whose graphs are star or path. We also give a sufficient condition for sign pattern matrices whose graphs are trees to allow diagonalizability. Further, we give a necessary condition for a sign pattern matrix to require diagonalizability and characterize all star sign pattern matrices that require diagonalizability.

On the Eventual Exponential Positivity of Some Tree Sign Patterns

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.

Potentially stable and 5-by-5 spectrally arbitrary tree sign pattern matrices with all edges negative

Characterization of potentially stable sign pattern matrices has been a long-standing open problem. In this paper, we give some sufficient conditions for tree sign pattern matrices with all edges negative to allow a properly signed nest. We also characterize potentially stable star and path sign pattern matrices with all edges negative. We give a conjecture on characterizing potentially stable tree sign pattern matrices with all edges negative in terms of allowing a properly signed nest which is verified to be true for sign pattern matrices up to order 6. Finally, we characterize all 5-by-5 spectrally arbitrary tree sign pattern matrices with all edges negative.

Potentially Eventually Positive 2-generalized Star Sign Patterns

A sign pattern is a matrix whose entries belong to the set $\{+, -, 0\}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is said to be potentially eventually positive if there exists at least one real matrix $A$ with the same sign pattern as $\mathcal{A}$ and a positive integer $k_{0}$ such that $A^{k}>0$ for all $k\geq k_{0}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is said to be potentially eventually exponentially positive if there exists at least one real matrix $A$ with the same sign pattern as $\mathcal{A}$ and a nonnegative integer $t_{0}$ such that $e^{tA}=\sum_{k=0}^{\infty}\frac{t^{k}A^{k}}{k!}>0$ for all $t\geq t_{0}$. Identifying necessary and sufficient conditions for an $n$-by-$n$ sign pattern to be potentially eventually positive (respectively, potentially eventually exponentially positive), and classifying these sign patterns are open problems. In this article, the potential eventual positivity of the $2$-generalized star sign patterns is investigated. All the minimal potentially eventually positive $2$-generalized star sign patterns are identified. Consequently, all the potentially eventually positive $2$-generalized star sign patterns are classified. As an application, all the minimal potentially eventually exponentially positive $2$-generalized star sign patterns are identified. Consequently, all the potentially eventually exponentially positive $2$-generalized star sign patterns are classified.

On some sign patterns of algebraically positive matrices

The concept of an algebraically positive matrix was introduced by Kirkland, Qiao, and Zhan in 2016. A real square matrix A is said to be algebraically positive if there exists a real polynomial f such that f(A) is a positive matrix. In this paper, we give a new characterization of algebraically positive matrices, and characterize all tree sign pattern matrices that allow algebraic positivity, and all star and path sign pattern matrices that require algebraic positivity. Also, all tree sign pattern matrices of order less than 6 requiring algebraic positivity are characterized.

Characterization of star sign patterns that require [formula omitted

The refined inertia of a square real matrix B, denoted ri(B), is the ordered 4-tuple (n+(B),n−(B),nz(B),2np(B)), where n+(B) (resp., n−(B)) is the number of eigenvalues of B with positive (resp., negative) real part, nz(B) is the number of zero eigenvalues of B, and 2np(B) is the number of pure imaginary eigenvalues of B. For n≥3, 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. An n×n sign pattern A is said to require Hn if Hn={ri(B)|B∈Q(A)}. The star sign patterns of order n≥5 that require Hn are characterized. More specifically, it is shown that for each n≥5, a star sign pattern requires Hn if and only if it is equivalent to one of the five sign patterns identified in the paper.

Potentially eventually positive star sign patterns

An $n$-by-$n$ real matrix $A$ is eventually positive if there exists a positive integer $k_{0}$ such that $A^{k}>0$ for all $k\geq k_{0}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is potentially eventually positive (PEP) if there exists an eventually positive real matrix $A$ with the same sign pattern as $\mathcal{A}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is a minimal potentially eventually positive sign pattern (MPEP sign pattern) if $\mathcal{A}$ is PEP and no proper subpattern of $\mathcal{A}$ is PEP. Berman, Catral, Dealba, et al. [Sign patterns that allow eventual positivity, {\it ELA}, 19(2010): 108-120] established some sufficient and some necessary conditions for an $n$-by-$n$ sign pattern to allow eventual positivity and classified the potentially eventually positive sign patterns of order $n\leq 3$. However, the identification and classification of PEP signpatterns of order $n\geq 4$ remain open. In this paper, all the $n$-by-$n$ PEP star sign patterns are classified by identifying all the MPEP star sign patterns.

Refined inertias of tree sign-patterns

The refined inertia (n+,n ,nz,2np) of a real matrix is the ordered 4-tuple that subdivides the number n0 of eigenvalues with zero real part in the inertia (n+,n ,n0) 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 s pattern (necessarily a path or a star) to require H4. 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.

Potentially eventually positive double star sign patterns

In this work, the potentially eventually positive double star sign patterns are characterized. It is shown that, up to equivalence, an ( m + n ) × ( m + n ) double star sign pattern is potentially eventually positive if and only if it is a superpattern of four specific sign patterns.

On potentially nilpotent double star sign patterns

A matrix Open image in new window whose entries come from the set {+, −, 0} is called a sign pattern matrix, or sign pattern. A sign pattern is said to be potentially nilpotent if it has a nilpotent realization. In this paper, the characterization problem for some potentially nilpotent double star sign patterns is discussed. A class of double star sign patterns, denoted by \( \mathcal{D}\mathcal{S}\mathcal{S}\mathcal{P} \)(m, 2), is introduced. We determine all potentially nilpotent sign patterns in \( \mathcal{D}\mathcal{S}\mathcal{S}\mathcal{P} \)(3, 2) and \( \mathcal{D}\mathcal{S}\mathcal{S}\mathcal{P} \)(5, 2), and prove that one sign pattern in \( \mathcal{D}\mathcal{S}\mathcal{S}\mathcal{P} \)(3, 2) is potentially stable.

Inertia sets of symmetric 2-generalized star sign patterns

A sign pattern is a matrix whose entries are from the set {+−0}. Associated with each sign pattern A is a qualitative class of matrices Q(A). For a symmetric sign pattern A of order n, the inertia set of A is the set , and the symmetric inertia set , where i(B) is the inertia of B. In this article we characterize the symmetric 2-generalized star sign patterns that require unique (symmetric) inertia, and also give the (symmetric) inertia set of a symmetric 2-generalized star sign pattern.

Spectrally arbitrary star sign patterns

An n × n sign pattern S n is spectrally arbitrary if, for any given real monic polynomial g( x) of degree n, there is a real matrix having sign pattern S n and characteristic polynomial g( x). All n × n star sign patterns that are spectrally arbitrary, and all minimal such patterns, are characterized. This subsequently leads to an explicit characterization of all n × n star sign patterns that are potentially nilpotent. It is shown that any super-pattern of a spectrally arbitrary star sign pattern is also spectrally arbitrary.

Inertia sets of two classes of symmetric sign patterns*1

A sign pattern is a matrix whose entries are from the set {+,−,0}. For a symmetric sign pattern A of order n , the inertia set of A is the set of inertias of all real symmetric matrices with the same sign pattern as A . The purpose of this paper is to characterize the inertia sets of symmetric star and nonnegative symmetric tridiagonal sign patterns.

Inertia sets of two classes of symmetric sign patterns

A sign pattern is a matrix whose entries are from the set {+,−,0}. For a symmetric sign pattern A of order n, the inertia set of A is the set of inertias of all real symmetric matrices with the same sign pattern as A. The purpose of this paper is to characterize the inertia sets of symmetric star and nonnegative symmetric tridiagonal sign patterns.

Two classes of symmetric sign patterns that require unique inertia

A sign pattern is a matrix whose entries are from the set {+,−,0}. Associated with each sign pattern A of order n is a qualitative class of A,defined by Q(A). For a symmetric sign pattern A of order n,the inertia of A is a set i(A)={i(B)=(i +(B),i −(B),i 0(B)) | B=B T H∈Q(A)}, where i + (B) (respectively,i − (B),i 0(B) denotes the number of positive (respectively, negative, zero) eigenvalues. That the symmetric sign pattern A requires unique intertia means i(B 1)=i(B 2) for all real symmetric matrices B 1,B 2∈Q(A). The purpose of this paper is to characterize double star and cycle sign patterns that require unique inertia. Further, their unique inertia is also obtained.

On the potential stability of star sign pattern matrices

A sign pattern matrix A is called potentially stable if there exists a real matrix B∈Q(A) such that B is stable. A combinatorially symmetric sign pattern matrix A is called tree (star) sign pattern if its graph G(A) is a tree (star). In this paper, a characterization for a star sign pattern matrix being potentially stable is presented.

Sign pattern matrices that allow a nilpotent matrix

We characterise some star sign pattern matrices and linear tree sign pattern matrices that allow a nilpotent matrix.