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 inhabiting the left half-plane that are universally realizable

2021 ◽  
Vol 10 (1) ◽  
pp. 180-192
Author(s):  
Ricardo L. Soto
Keyword(s):  
Canonical Form ◽  
Half Plane ◽  
New Left ◽  
Nonnegative Matrix ◽  
Complex Numbers ◽  
Jordan Canonical Form ◽  
Positivity Condition ◽  
Left Half ◽  
Left Half Plane ◽  
New Criteria

Abstract Let Λ = {λ1, λ2, . . ., λ n } be a list of complex numbers. Λ is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. Minc ([21],1981) showed that if Λ is diagonalizably positively realizable, then Λ is universally realizable. The positivity condition is essential for the proof of Minc, and the question whether the result holds for nonnegative realizations has been open for almost forty years. Recently, two extensions of the Minc’s result have been proved in ([5], 2018) and ([12], 2020). In this work we characterize new left half-plane lists (λ1 > 0, Re λ i ≤ 0, i = 2, . . ., n) no positively realizable, which are universally realizable. We also show new criteria which allow to decide about the universal realizability of more general lists, extending in this way some previous results.

scholarly journals Permutative universal realizability

2021 ◽  
Vol 9 (1) ◽  
pp. 66-77
Author(s):  
Ricardo L. Soto ◽  
Ana I. Julio ◽  
Jaime H. Alfaro
Keyword(s):  
Canonical Form ◽  
Nonnegative Matrix ◽  
Positive Element ◽  
Complex Numbers ◽  
Sufficient Condition ◽  
Real Numbers ◽  
Jordan Canonical Form

Abstract A list of complex numbers Λ is said to be realizable, if it is the spectrum of a nonnegative matrix. In this paper we provide a new sufficient condition for a given list Λ to be universally realizable (UR), that is, realizable for each possible Jordan canonical form allowed by Λ. Furthermore, the resulting matrix (that is explicity provided) is permutative, meaning that each of its rows is a permutation of the first row. In particular, we show that a real Suleĭmanova spectrum, that is, a list of real numbers having exactly one positive element, is UR by a permutative 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 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 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 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 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.

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