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.

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 The inverse eigenvalue problem for real eventually positive matrices

Filomat ◽  
2007 ◽  
Vol 21 (1) ◽  
pp. 1-16
Author(s):  
Jianbiao Chen ◽  
Zhaoliang Xu
Keyword(s):  
Eigenvalue Problem ◽  
Inverse Eigenvalue Problem ◽  
Constructive Method ◽  
Complex Numbers ◽  
Positive Matrix ◽  
Complete Answer ◽  
Positive Matrices ◽  
Inverse Eigenvalue ◽  
The Right

The eventually positive inverse eigenvalue problem(EPIEP) asks when a list ? = (???2,???n) of n complex numbers is the spectrum of a real n ? n eventually positive matrix. In this paper, a constructive method is presented to solve the EPIEP. Using this method, it is easy to find the realizing matrix, i.e. a real eventually positive matrix with the spectrum ?. And a complete answer to the EPIEP prescribed the left and the right Perron eigenvectors of the realizing matrix is also given.

scholarly journals On the inverse eigenvalue problem of symmetric nonnegative matrices

2019 ◽  
Vol 14 (1) ◽  
pp. 11-19
Author(s):  
A. M. Nazari ◽  
A. Mashayekhi ◽  
A. Nezami
Keyword(s):  
Eigenvalue Problem ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Nonnegative Matrices ◽  
Real Numbers ◽  
Inverse Eigenvalue ◽  
The Given

AbstractIn this paper, at first for a given set of real numbers with only one positive number, and in continue for a given set of real numbers in special conditions, we construct a symmetric nonnegative matrix such that the given set is its spectrum.

scholarly journals Brauer's theorem and nonnegative matrices with prescribed diagonal entries

2019 ◽  
Vol 35 ◽  
pp. 53-64 ◽  
Author(s):  
Ricardo Soto ◽  
Ana Julio ◽  
Macarena Collao
Keyword(s):  
Inverse Problem ◽  
Eigenvalue Problem ◽  
Inverse Eigenvalue Problem ◽  
Complex Numbers ◽  
Nonnegative Matrices ◽  
Simple Condition ◽  
Inverse Spectral Problems ◽  
Real Numbers ◽  
Inverse Eigenvalue ◽  
Necessary And Sufficient

The problem of the existence and construction of nonnegative matrices with prescribed eigenvalues and diagonal entries is an important inverse problem, interesting by itself, but also necessary to apply a perturbation result, which has played an important role in the study of certain nonnegative inverse spectral problems. A number of partial results about the problem have been published by several authors, mainly by H. \v{S}migoc. In this paper, the relevance of a Brauer's result, and its implication for the nonnegative inverse eigenvalue problem with prescribed diagonal entries is emphasized. As a consequence, given a list of complex numbers of \v{S}migoc type, or a list $\Lambda = \left\{\lambda _{1},\ldots ,\lambda _{n} \right \}$ with $\operatorname{Re}\lambda _{i}\leq 0,$ $\lambda _{1}\geq -\sum\limits_{i=2}^{n}\lambda _{i}$, and $\left\{-\sum\limits_{i=2}^{n}\lambda _{i},\lambda _{2},\ldots ,\lambda _{n} \right\}$ being realizable; and given a list of nonnegative real numbers $% \Gamma = \left\{\gamma _{1},\ldots ,\gamma _{n} \right\}$, the remarkably simple condition $\gamma _{1}+\cdots +\gamma _{n} = \lambda _{1}+\cdots +\lambda _{n}$ is necessary and sufficient for the existence and construction of a realizing matrix with diagonal entries $\Gamma .$ Conditions for more general lists of complex numbers are also given.

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 Solutions of a Quadratic Inverse Eigenvalue Problem for Damped Gyroscopic Second-Order Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Hong-Xiu Zhong ◽  
Guo-Liang Chen ◽  
Xiang-Yun Zhang
Keyword(s):  
Eigenvalue Problem ◽  
Inverse Eigenvalue Problem ◽  
Complex Numbers ◽  
Quadratic Eigenvalue Problem ◽  
Quadratic Pencil ◽  
Inverse Eigenvalue ◽  
Particular Solution ◽  
Second Order Systems ◽  
Quadratic Eigenvalue ◽  
The Given

Givenkpairs of complex numbers and vectors (closed under conjugation), we consider the inverse quadratic eigenvalue problem of constructingn×nreal matricesM,D,G, andK, whereM>0,KandDare symmetric, andGis skew-symmetric, so that the quadratic pencilQ(λ)=λ2M+λ(D+G)+Khas the givenkpairs as eigenpairs. First, we construct a general solution to this problem withk≤n. Then, with the special propertiesD=0andK<0, we construct a particular solution. Numerical results illustrate these solutions.

A note on the real nonnegative inverse eigenvalue problem

2016 ◽  
Vol 31 ◽  
pp. 765-773 ◽  
Author(s):  
Raphael Loewy
Keyword(s):  
Eigenvalue Problem ◽  
Special Form ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Real Numbers ◽  
The Real ◽  
Direct Sum ◽  
Inverse Eigenvalue

The Real Nonnegative Inverse Eigenvalue Problem (RNIEP) asks when is a list \[ \sigma=(\lambda_1, \lambda_2,\ldots,\lambda_n)\] consisting of real numbers the spectrum of an $n \times n$ nonnegative matrix $A$. In that case, $\sigma$ is said to be realizable and $A$ is a realizing matrix. In a recent paper dealing with RNIEP, P.~Paparella considered cases of realizable spectra where a realizing matrix can be taken to have a special form, more precisely such that the entries of each row are obtained by permuting the entries of the first row. A matrix of this form is called permutative. Paparella raised the question whether any realizable list $\sigma$ can be realized by a permutative matrix or a direct sum of permutative matrices. In this paper, it is shown that in general the answer is no.

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