Sign patterns of rational matrices with large rank II

10.13001/1081-3810.3027 ◽

2015 ◽

Vol 30 ◽

Cited By ~ 1

Author(s):

Craig Erickson

Keyword(s):

Sign Pattern ◽

Sign Patterns ◽

Necessary And Sufficient

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 (nâ1)(nâ2)/2 + 2

Potentially eventually positive star sign patterns

10.13001/1081-3810.2963 ◽

2016 ◽

Vol 31 ◽

pp. 541-548

Author(s):

Yu Ber-Lin ◽

Huang Ting-Zhu ◽

Jie Cui ◽

Deng Chunhua

Keyword(s):

Positive Integer ◽

Necessary Conditions ◽

Positive Sign ◽

Real Matrix ◽

Sign Pattern ◽

Positive Real ◽

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.

On the infinite Borwein product raised to a positive real power

The Ramanujan Journal ◽

10.1007/s11139-021-00519-3 ◽

2021 ◽

Author(s):

Michael J. Schlosser ◽

Nian Hong Zhou

Keyword(s):

Asymptotic Formula ◽

Circle Method ◽

Sign Pattern ◽

Real Numbers ◽

Positive Real ◽

Infinite Products ◽

Real Power ◽

Sign Patterns ◽

Divisibility Properties ◽

Made In

AbstractIn this paper, we study properties of the coefficients appearing in the q-series expansion of $$\prod _{n\ge 1}[(1-q^n)/(1-q^{pn})]^\delta $$ ∏ n ≥ 1 [ ( 1 - q n ) / ( 1 - q pn ) ] δ , the infinite Borwein product for an arbitrary prime p, raised to an arbitrary positive real power $$\delta $$ δ . We use the Hardy–Ramanujan–Rademacher circle method to give an asymptotic formula for the coefficients. For $$p=3$$ p = 3 we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent $$\delta $$ δ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the $$p=3$$ p = 3 case.

Inertia sets allowed by matrix patterns

10.13001/1081-3810.3647 ◽

2018 ◽

Vol 34 ◽

pp. 343-355 ◽

Cited By ~ 1

Author(s):

Adam Berliner ◽

Dale Olesky ◽

Pauline Van den Driessche

Keyword(s):

Dynamical Systems ◽

Sufficient Condition ◽

Sign Pattern ◽

Zero Eigenvalue ◽

Motivated by the possible onset of instability in dynamical systems associated with a zero eigenvalue, sets of inertias $\sn_n$ and $\SN{n}$ for sign and zero-nonzero patterns, respectively, are introduced. For an $n\times n$ sign pattern $\mc{A}$ that allows inertia $(0,n-1,1)$, a sufficient condition is given for $\mc{A}$ and every superpattern of $\mc{A}$ to allow $\sn_n$, and a family of such irreducible sign patterns for all $n\geq 3$ is specified. All zero-nonzero patterns (up to equivalence) that allow $\SN{3}$ and $\SN{4}$ are determined, and are described by their associated digraphs.

Sign patterns of rational matrices with large rank

European Journal of Combinatorics ◽

10.1016/j.ejc.2014.06.001 ◽

2014 ◽

Vol 42 ◽

pp. 107-111 ◽

Cited By ~ 3

Author(s):

Yaroslav Shitov

Keyword(s):

Large Rank ◽

Spectral Properties of Sign Patterns

10.13001/ela.2020.5057 ◽

2020 ◽

Vol 36 (36) ◽

pp. 183-197

Author(s):

Michael Cavers ◽

Jonathan Fischer ◽

Kevin N. Vander Meulen

Keyword(s):

Spectral Properties ◽

Infinite Family ◽

Sign Pattern ◽

Sign Patterns ◽

Spectrally Arbitrary

In this paper, an infinite family of irreducible sign patterns that are spectrally arbitrary, for which the nilpotent-Jacobian method does not apply, is given. It is demonstrated that it is possible for an irreducible sign pattern to be refined inertially arbitrary and not spectrally arbitrary. It is observed that not every nonzero spectrally arbitrary pattern has a signing which is spectrally arbitrary. It is also shown that every superpattern of the reducible pattern $\T_2 \oplus \T_2$ is spectrally arbitrary.

From real to complex sign pattern matrices

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700031518 ◽

1998 ◽

Vol 57 (1) ◽

pp. 159-172 ◽

Cited By ~ 13

Author(s):

Carolyn A. Eschenbach ◽

Frank J. Hall ◽

Zhongshan Li

Keyword(s):

Matrix Analysis ◽

Complex Case ◽

Sign Pattern ◽

Pattern Class ◽

Class A ◽

Open Questions ◽

Sign Patterns ◽

Complex Patterns ◽

Ray Patterns ◽

Real Matrices

This paper extends some fundamental concepts of qualitative matrix analysis from sign pattern classes of real matrices to sign pattern classes of complex matrices. A complex sign pattern and its corresponding sign pattern class are defined in such a way that they generalize the definitions of a (real) sign pattern and its corresponding sign pattern class. A survey of several qualitative results on complex sign patterns is presented. In particular, sign nonsingular complex patterns are investigated. The type of region in the complex plane representing the distribution of the determinants of the matrices in the sign pattern class of a sign nonsingular complex pattern is identified. Cyclically nonnegative complex patterns and complex patterns that are signature similar to nonnegative patterns are characterized. Extensions of sign stable and sign semistable patterns from the real to the complex case are given. Results on ray patterns are also obtained. Finally, many open questions are mentioned.

Potentially Eventually Positive 2-generalized Star Sign Patterns

10.13001/1081-3810.3876 ◽

2019 ◽

Vol 35 ◽

pp. 100-115

Author(s):

Yu Ber-Lin ◽

Ting-Zhu Huang ◽

Xu Sanzhang

Keyword(s):

Positive Integer ◽

Nonnegative Integer ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Real Matrix ◽

Sign Pattern ◽

Open Problems ◽

Sign Patterns ◽

Necessary And Sufficient

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 the Eventual Exponential Positivity of Some Tree Sign Patterns

Symmetry ◽

10.3390/sym13091669 ◽

2021 ◽

Vol 13 (9) ◽

pp. 1669

Author(s):

Ber-Lin Yu ◽

Zhongshan Li ◽

Sanzhang Xu

Keyword(s):

Open Problem ◽

Double Star ◽

Real Matrix ◽

Sign Pattern ◽

Preliminary Results ◽

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.

Essential sign change numbers of full sign pattern matrices

Special Matrices ◽

10.1515/spma-2016-0023 ◽

2016 ◽

Vol 4 (1) ◽

Author(s):

Xiaofeng Chen ◽

Wei Fang ◽

Wei Gao ◽

Yubin Gao ◽

Guangming Jing ◽

...

Keyword(s):

Sign Pattern ◽

Minimum Rank ◽

Open Problems ◽

Sign Change ◽

Real Polynomials ◽

Sign Patterns ◽

Sign Vector ◽

Pattern Matrix ◽

Point Line ◽

Real Matrices

AbstractA sign pattern (matrix) is a matrix whose entries are from the set {+, −, 0} and a sign vector is a vector whose entries are from the set {+, −, 0}. A sign pattern or sign vector is full if it does not contain any zero entries. The minimum rank of a sign pattern matrix A is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of A. The notions of essential row sign change number and essential column sign change number are introduced for full sign patterns and condensed sign patterns. By inspecting the sign vectors realized by a list of real polynomials in one variable, a lower bound on the essential row and column sign change numbers is obtained. Using point-line confiurations on the plane, it is shown that even for full sign patterns with minimum rank 3, the essential row and column sign change numbers can differ greatly and can be much bigger than the minimum rank. Some open problems concerning square full sign patterns with large minimum ranks are discussed.