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 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.

On the Construction of Nonnegative 5×5 Matrices from Spectrum Data

2013 ◽  
Vol 444-445 ◽  
pp. 621-624
Author(s):  
Zhi Bing Liu ◽  
Zhen Tu ◽  
Cheng Feng Xu
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Nonnegative Matrices ◽  
Necessary And Sufficient ◽  
Spectrum Data

This paper studies the construction problems of five order nonnegative matrices from spectrum data. Let be a list of complex numbers with . Necessary and sufficient conditions for the existence of an entry-wise nonnegative 5×5 matrix with spectrum are presented.

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 A Note on NIEP for Leslie and Doubly Leslie Matrices

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 559
Author(s):  
Luis Medina ◽  
Hans Nina ◽  
Elvis Valero
Keyword(s):  
Inverse Problem ◽  
Sufficient Conditions ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Complex Numbers ◽  
The Matrix ◽  
Leslie Matrices ◽  
Inverse Eigenvalue ◽  
Necessary And Sufficient ◽  
Nonnegative Eigenvalue

The nonnegative inverse eigenvalue problem (NIEP) consists of finding necessary and sufficient conditions for the existence of a nonnegative matrix with a given list of complex numbers as its spectrum. If the matrix is required to be Leslie (doubly Leslie), the problem is called the Leslie (doubly Leslie) nonnegative eigenvalue inverse problem. In this paper, necessary and/or sufficient conditions for the existence and construction of Leslie and doubly Leslie matrices with a given spectrum are considered.

scholarly journals Centrosymmetric universal realizability

2021 ◽  
Vol 37 ◽  
pp. 680-691
Author(s):  
Ana Julio ◽  
Yankis R. Linares ◽  
Ricardo L. Soto
Keyword(s):  
Canonical Form ◽  
Sufficient Conditions ◽  
Nonnegative Matrix ◽  
Complex Numbers ◽  
Jordan Canonical Form ◽  
Centrosymmetric Matrix

A list $\Lambda =\{\lambda_{1},\ldots,\lambda_{n}\}$ of complex numbers is said to be realizable, if it is the spectrum of an entrywise nonnegative matrix $A$. In this case, $A$ is said to be a realizing matrix. $\Lambda$ is said to be universally realizable, if it is realizable for each possible Jordan canonical form (JCF) allowed by $\Lambda$. The problem of the universal realizability of spectra is called the universal realizability problem (URP). Here, we study the centrosymmetric URP, that is, the problem of finding a nonnegative centrosymmetric matrix for each JCF allowed by a given list $\Lambda $. In particular, sufficient conditions for the centrosymmetric URP to have a solution are generated.

scholarly journals The role of certain Brauer and Rado results in the nonnegative inverse spectral problems

2020 ◽  
Vol 36 (36) ◽  
pp. 484-502
Author(s):  
Ana Julio ◽  
Ricardo Soto
Keyword(s):  
Canonical Form ◽  
Sufficient Conditions ◽  
Nonnegative Matrix ◽  
Complex Numbers ◽  
Inverse Spectral Problems ◽  
Jordan Canonical Form ◽  
Spectral Problems ◽  
Inverse Spectral

It is said that a list $\Lambda =\{\lambda _{1},\ldots ,\lambda _{n}\}$ of complex numbers is realizable, if it is the spectrum of a nonnegative matrix $A$. It is said that $\Lambda $ is universally realizable if it is realizable for each possible Jordan canonical form allowed by $\Lambda$. This work does not contain new results. As its title says, its goal is to show and emphasize the relevance and importance of certain results, by Brauer and Rado, in the study of nonnegative inverse spectral problems. It is shown that virtually all known results, which give sufficient conditions for $\Lambda$ to be realizable or universally realizable, can be obtained from results by Brauer and Rado. Moreover, from these results, a realizing matrix may always be constructed.

scholarly journals Spectra universally realizable by doubly stochastic matrices

2018 ◽  
Vol 6 (1) ◽  
pp. 301-309 ◽  
Author(s):  
Macarena Collao ◽  
Mario Salas ◽  
Ricardo L. Soto
Keyword(s):  
Complete Solution ◽  
Sufficient Conditions ◽  
Nonnegative Matrix ◽  
Stochastic Matrices ◽  
Jordan Canonical Form ◽  
Doubly Stochastic ◽  
Elementary Divisors ◽  
Current State ◽  
Doubly Stochastic Matrices ◽  
Solution Matrix

Abstract A list of complex numbers Λ = { λ1, . . . , λn} is said to be realizable if it is the spectrum of an entrywise nonnegative matrix, and universally realizable if there exists a nonnegative matrix with spectrum Λ for each Jordan canonical form associated with Λ. The problem of characterizing the lists which are universally realizable is called the nonnegative inverse elementary divisors problem (NIEDP). This is a hard problem, which remains unsolved. A complete solution, if any, is still far from the current state of the art in the problem. In particular, in this paper we consider the NIEDP for generalized doubly stochastic matrices, and give new sufficient conditions for the existence and construction of a solution matrix. These conditions improve those given in [ELA 30 (2015) 704-720]

scholarly journals Toeplitz nonnegative realization of spectra via companion matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 230-245
Author(s):  
Macarena Collao ◽  
Mario Salas ◽  
Ricardo L. Soto
Keyword(s):  
Eigenvalue Problem ◽  
Half Plane ◽  
Toeplitz Matrix ◽  
Toeplitz Matrices ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Complex Numbers ◽  
Inverse Eigenvalue ◽  
Companion Matrices ◽  
Solution Matrix

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

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.

scholarly journals The multipliers of multiple trigonometric Fourier series

2016 ◽  
Vol 6 (1) ◽  
Author(s):  
Aizhan Ydyrys ◽  
Lyazzat Sarybekova ◽  
Nazerke Tleukhanova
Keyword(s):  
Fourier Series ◽  
Sufficient Conditions ◽  
Regular System ◽  
Multiple Fourier Series ◽  
Trigonometric Fourier Series ◽  
Lorentz Spaces ◽  
Complex Numbers ◽  
Multiple Trigonometric Fourier Series

Abstract We study the multipliers of multiple Fourier series for a regular system on anisotropic Lorentz spaces. In particular, the sufficient conditions for a sequence of complex numbers {λk}k∈Zn in order to make it a multiplier of multiple trigonometric Fourier series from Lp[0; 1]n to Lq[0; 1]n , p > q. These conditions include conditions Lizorkin theorem on multipliers.

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