scholarly journals ON NONNEGATIVE MATRICES WITH A FULLY CYCLIC PERIPHERAL SPECTRUM

1990 ◽  
Vol 21 (1) ◽  
pp. 65-70
Author(s):  
Bit-Shun Tam
Keyword(s):  
Spectral Radius ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Nonnegative Matrix ◽  
Nonnegative Matrices ◽  
Complex Matrix ◽  
Peripheral Spectrum ◽  
Rho A ◽  
Square Complex

Let $A$ be a square complex matrix. We denote by $\rho(A)$ the spectral radius of $A$. The set of eigenvalues of $A$ with modulus $\rho(A)$ is called the peripheral spectrum of $A$. The latter set is said to to be fully cyclic if whenever $\rho(A)\alpha x =Ax$, $x\neq 0$, $|a|= 1$, then $|x|(sgn \ x)^k$ is an eigenvector of $A$ corresponding to $\rho(A)\alpha^k$ for all integers $k$. In this paper we give some necessary conditions and a set of sufficient conditions for a nonnegative matrix to have a fully cyclic peripheral spectrum. Our conditions are given in terms of the reduced graph of a nonnegative matrix.

scholarly journals Optimal Gersgorin-style estimation of the largest singular value. II

2016 ◽  
Vol 31 ◽  
pp. 679-685
Author(s):  
Charles Johnson ◽  
J. Pena ◽  
Tomasz Szulc
Keyword(s):  
Spectral Radius ◽  
Linear Algebra ◽  
Nonnegative Matrix ◽  
Singular Value ◽  
Nonnegative Matrices ◽  
Electronic Journal ◽  
Complex Matrix ◽  
Class Of Matrices

In estimating the largest singular value in the class of matrices equiradial with a given $n$-by-$n$ complex matrix $A$, it was proved that it is attained at one of $n(n-1)$ sparse nonnegative matrices (see C.R.~Johnson, J.M.~Pe{\~n}a and T.~Szulc, Optimal Gersgorin-style estimation of the largest singular value; {\em Electronic Journal of Linear Algebra Algebra Appl.}, 25:48--59, 2011). Next, some circumstances were identified under which the set of possible optimizers of the largest singular value can be further narrowed (see C.R.~Johnson, T.~Szulc and D.~Wojtera-Tyrakowska, Optimal Gersgorin-style estimation of the largest singular value, {\it Electronic Journal of Linear Algebra Algebra Appl.}, 25:48--59, 2011). Here the cardinality of the mentioned set for $n$-by-$n$ matrices is further reduced. It is shown that the largest singular value, in the class of matrices equiradial with a given $n$-by-$n$ complex matrix, is attained at one of $n(n-1)/2$ sparse nonnegative matrices. Finally, an inequality between the spectral radius of a $3$-by-$3$ nonnegative matrix $X$ and the spectral radius of a modification of $X$ is also proposed.

scholarly journals On the Perron-Frobenius Theory of Mv-matrices and equivalent properties to eventually exponentially nonnegative matrices

2019 ◽  
Vol 35 ◽  
pp. 424-440
Author(s):  
Thaniporn Chaysri ◽  
Dimitrios Noutsos
Keyword(s):  
Necessary Conditions ◽  
Nonnegative Matrix ◽  
Nonnegative Matrices ◽  
Numerical Examples ◽  
Generalized Eigenvectors ◽  
Sufficient And Necessary Conditions ◽  
Left And Right ◽  
Equivalent Properties

Mv−matrix is a matrix of the form A = sI −B, where 0 ≤ ρ(B) ≤ s and B is an eventually nonnegative matrix. In this paper, Mv−matrices concerning the Perron-Frobenius theory are studied. Specifically, sufficient and necessary conditions for an Mv−matrix to have positive left and right eigenvectors corresponding to its eigenvalue with smallest real part without considering or not if index0B ≤ 1 are stated and proven. Moreover, analogous conditions for eventually nonnegative matrices or Mv−matrices to have all the non Perron eigenvectors or generalized eigenvectors not being nonnegative are studied. Then, equivalent properties of eventually exponentially nonnegative matrices and Mv−matrices are presented.  Various numerical examples are given to support our theoretical findings.

scholarly journals Sharp Bounds on the Spectral Radius of Nonnegative Matrices and Comparison to the Frobenius’ Bounds

2020 ◽  
Vol 14 ◽  
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Discrete Time ◽  
Upper Bound ◽  
Nonnegative Matrix ◽  
Nonnegative Matrices ◽  
Sharp Bounds ◽  
Similarity Transformations ◽  
Linear Invariant ◽  
The Stability

In this paper, a new upper bound and a new lower bound for the spectral radius of a nοnnegative matrix are proved by using similarity transformations. These bounds depend only on the elements of the nonnegative matrix and its row sums and are compared to the well-established upper and lower Frobenius’ bounds. The proposed bounds are always sharper or equal to the Frobenius’ bounds. The conditions under which the new bounds are sharper than the Frobenius' ones are determined. Illustrative examples are also provided in order to highlight the sharpness of the proposed bounds in comparison with the Frobenius’ bounds. An application to linear invariant discrete-time nonnegative systems is given and the stability of the systems is investigated. The proposed bounds are computed with complexity O(n2).

scholarly journals Nonnegative Inverse Elementary Divisors Problem for Lists with Nonnegative Real Parts

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1662
Author(s):  
Hans Nina ◽  
Hector Flores Callisaya ◽  
H. Pickmann-Soto ◽  
Jonnathan Rodriguez
Keyword(s):  
Sufficient Conditions ◽  
Nonnegative Matrix ◽  
Complex Numbers ◽  
Nonnegative Matrices ◽  
Elementary Divisors ◽  
Complex Eigenvalues

In this paper, sufficient conditions for the existence and construction of nonnegative matrices with prescribed elementary divisors for a list of complex numbers with nonnegative real part are obtained, and the corresponding nonnegative matrices are constructed. In addition, results of how to perturb complex eigenvalues of a nonnegative matrix while keeping its elementary divisors and its nonnegativity are derived.

scholarly journals Note on the Jordan form of an irreducible eventually nonnegative matrix

2015 ◽  
Vol 30 ◽  
Author(s):  
Leslie Hogben ◽  
Bit-Shun Tam ◽  
Ulrica Wilson
Keyword(s):  
Positive Integer ◽  
Nonnegative Matrix ◽  
Positive Answer ◽  
Jordan Form ◽  
Complex Matrix ◽  
Open Question ◽  
Square Complex

A square complex matrix A is eventually nonnegative if there exists a positive integer k_0 such that for all k ≥ k_0, A^k ≥ 0; A is strongly eventually nonnegative if it is eventually nonnegative and has an irreducible nonnegative power. It is proved that a collection of elementary Jordan blocks is a Frobenius Jordan multiset with cyclic index r if and only if it is the multiset of elementary Jordan blocks of a strongly eventually nonnegative matrix with cyclic index r. A positive answer to an open question and a counterexample to a conjecture raised by Zaslavsky and Tam are given. It is also shown that for a square complex matrix A with index at most one, A is irreducible and eventually nonnegative if and only if A is strongly eventually nonnegative.

scholarly journals The inverse eigenvalue problem for Leslie matrices

2019 ◽  
Vol 35 ◽  
pp. 319-330 ◽  
Author(s):  
Luca Benvenuti
Keyword(s):  
Eigenvalue Problem ◽  
Sufficient Conditions ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Leslie Matrix ◽  
Complex Numbers ◽  
Nonnegative Matrices ◽  
Minimal Dimension ◽  
Leslie Matrices ◽  
Inverse Eigenvalue

The Nonnegative Inverse Eigenvalue Problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of $n$ complex numbers to be the spectrum of an entry--wise nonnegative matrix of dimension $n$. This is a very difficult and long standing problem and has been solved only for $n\leq 4$. In this paper, the NIEP for a particular class of nonnegative matrices, namely Leslie matrices, is considered. Leslie matrices are nonnegative matrices, with a special zero--pattern, arising in the Leslie model, one of the best known and widely used models to describe the growth of populations. The lists of nonzero complex numbers that are subsets of the spectra of Leslie matrices are fully characterized. Moreover, the minimal dimension of a Leslie matrix having a given list of three numbers among its spectrum is provided. This result is partially extended to the case of lists of $n > 2$ real numbers.

scholarly journals Noetherian properties in composite generalized power series rings

2020 ◽  
Vol 18 (1) ◽  
pp. 1540-1551
Author(s):  
Jung Wook Lim ◽  
Dong Yeol Oh
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generalized Power Series ◽  
Ordered Monoid ◽  
Power Series Rings ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

Abstract Let ({\mathrm{\Gamma}},\le ) be a strictly ordered monoid, and let {{\mathrm{\Gamma}}}^{\ast }\left={\mathrm{\Gamma}}\backslash \{0\} . Let D\subseteq E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. Set \begin{array}{l}D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {E}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(0)\in D\right\}\hspace{.5em}\text{and}\\ \hspace{0.2em}D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {D}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(\alpha )\in I,\hspace{.5em}\text{for}\hspace{.25em}\text{all}\hspace{.5em}\alpha \in {{\mathrm{\Gamma}}}^{\ast }\right\}.\end{array} In this paper, we give necessary conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively ordered, and sufficient conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. Moreover, we give a necessary and sufficient condition for the ring D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. As corollaries, we give equivalent conditions for the rings D+({X}_{1},\ldots ,{X}_{n})E{[}{X}_{1},\ldots ,{X}_{n}] and D+({X}_{1},\ldots ,{X}_{n})I{[}{X}_{1},\ldots ,{X}_{n}] to be Noetherian.

scholarly journals On the equivalence of three-dimensional differential systems

2020 ◽  
Vol 18 (1) ◽  
pp. 1164-1172
Author(s):  
Jian Zhou ◽  
Shiyin Zhao
Keyword(s):  
Periodic Solutions ◽  
Differential System ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Systems ◽  
Structural Form ◽  
Differential Systems

Abstract In this paper, firstly, we study the structural form of reflective integral for a given system. Then the sufficient conditions are obtained to ensure there exists the reflective integral with these structured form for such system. Secondly, we discuss the necessary conditions for the equivalence of such systems and a general three-dimensional differential system. And then, we apply the obtained results to the study of the behavior of their periodic solutions when such systems are periodic systems in t.

Parallel Nonnegative Matrix Factorization via Newton Iteration

2016 ◽  
Vol 26 (03) ◽  
pp. 1650014 ◽  
Author(s):  
Markus Flatz ◽  
Marián Vajteršic
Keyword(s):  
Shared Memory ◽  
Matrix Factorization ◽  
Message Passing ◽  
Nonnegative Matrix Factorization ◽  
Nonnegative Matrix ◽  
Newton Iteration ◽  
Parallel Execution ◽  
Kkt Conditions ◽  
Nonnegative Matrices ◽  
First Order

The goal of Nonnegative Matrix Factorization (NMF) is to represent a large nonnegative matrix in an approximate way as a product of two significantly smaller nonnegative matrices. This paper shows in detail how an NMF algorithm based on Newton iteration can be derived using the general Karush-Kuhn-Tucker (KKT) conditions for first-order optimality. This algorithm is suited for parallel execution on systems with shared memory and also with message passing. Both versions were implemented and tested, delivering satisfactory speedup results.

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