Nonnegative matrices with prescribed spectrum and elementary divisors

Soto, Ricardo L.; Díaz, Roberto; Nina, Hans; Salas, Mario

doi:10.1016/j.laa.2013.09.034

Cited by 26 publications

(15 citation statements)

References 12 publications

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“…Then W T U = I, and We can also solve the inverse elementary divisors problem for a family of nonnegative matrices with constant row sums and spectrum Λ = {λ 1 , ..., λ n } in the following cases, which have been solved for a single nonnegative matrix in [13,15,3]:…”

Section: Inverse Elementary Divisors Problemmentioning

confidence: 99%

“…For each one of these cases, we may apply the same techniques used in [13,15,3] for a single nonnegative matrix. For example, in the first case i) Λ = {5, 3, 3, 3}, we use from [8], the Perfect matrix…”

Section: Inverse Elementary Divisors Problemmentioning

confidence: 99%

“…Closer to our approach in this paper is Rado's extension of Brauer's result, Theorem 2.2 below, which was first used by Perfect in [9] and later in [11] to derive sufficient conditions for the NIEP to have a solution. Both, Brauer and Rado results, have been also used to derive efficient sufficient conditions for the real and symmetric nonnegative inverse eigenvalue problem [11,12], as well as for the nonnegative inverse elementary divisors problem (NIEDP) [3,15,16]. In this paper we use the Rado result (Theorem 2.1 below) in a more general context, to construct in Section 2, a family of nonnegative matrices with prescribed spectrum.…”

Section: Introductionmentioning

confidence: 99%

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A family of nonnegative matrices with prescribed spectrum and elementary divisors

Soto¹,

Valero²

2016

imf

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A perturbation result, due to Rado, shows how to modify r eigenvalues of a matrix of order n, via a perturbation of rank r ≤ n, without changing any of the n − r remaining eigenvalues. This result extended a previous one, due to Brauer, on perturbations of rank 1. Both results have been exploited in connection with the nonnegative inverse eigenvalue problem and the nonnegative inverse elementary divisors problem. In this paper, we use the Rado result from a more general point of view, constructing a family of matrices with prescribed spectrum and elementary divisors, generalizing previous results. We also apply our results to the nonnegative pole assigment problem.

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Section: Inverse Elementary Divisors Problemmentioning

confidence: 99%

Section: Inverse Elementary Divisors Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A family of nonnegative matrices with prescribed spectrum and elementary divisors

Soto¹,

Valero²

2016

imf

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“…A number of sufficient conditions or realizability criteria for the NIEP to have a solution are known in the literature regarding the problem (see [5] and the references therein). In contrast, only a few works are known about the NIEDP (see [3,4,[6][7][8][9][10][11][12]) According to Minc [2], the NIEDP looks for to give an answer to the question: which matrices are similar to a nonnegative matrix?…”

Section: Introductionmentioning

confidence: 99%

Nonnegative Inverse Elementary Divisors Problem

Soto¹

2016

Applied Linear Algebra in Action

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Inverse eigenvalue problems appear in a wide variety of areas in the pure and applied mathematics. They have to do with the construction of a certain matrix from some spectral information. "ssociated with any inverse eigenvalue problem, there are two important issues the existence of a solution and the construction of a solution matrix. The purpose of this chapter is to study the nonneμative inverse elementary divisors problem hereafter, NIEDP and its state of the art. The elementary divisors of a given matrix A are the characteristic polynomials of the Jordan blocks of the Jordan canonical form of A. The NIEDP looks for necessary and sufficient conditions for the existence of a nonnegative matrix with prescribed elementary divisors. Most of the content of this chapter is based on recent results published by the author and collaborators from the Mathematics Department at Universidad Católica del Norte, Chile.Keywords: nonnegative matrices, elementary divisors, Jordan canonical form, nonnegative inverse eigenvalue problems, nonnegative inverse elementary divisors problems . IntroductionInverse problems appear in a wide variety of disciplines and they may be of many different kinds. Inverse eigenvalue problems, for instance, constitute an important subclass of inverse problems that arise in the context of mathematical modeling and parameter identification. " simple application of such problems is the construction of Leontief models in economics. Inverse eigenvalue problems have to do with constructing a certain matrix from some spectral information. "ssociated with any inverse eigenvalue problem, there are two important issues the existence of a solution and the construction of a solution matrix. The structure of the solution matrix usually it is not unique plays a fundamental role in the study of the inverse © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. eigenvalue problems. It is necessary to properly formulate the problem, otherwise it could become a trivial one. Chu and Golub [ ] say that an inverse eigenvalue problem should always be a structured problem. In this chapter, we study the Nonneμative Inverse Elementary Divisors Problem hereafter, the NIEDP , which is the problem of finding necessary and sufficient conditions for the existence of a nonnegative matrix with prescribed elementary divisors.Let A be an n × n complex matrix, and let ( ) In order that the problem be meaningful, the matrix A is required to have a particular structure. When A is required to be an entrywise nonnegative matrix, the problem is called the nonneμative inverse elementary divisors problem NIEDP see [ -] . The NIEDP is strongly related to another inverse problem, the nonneμative inverse eiμenvalue problem hereafter, the NIEP , which is the problem of determining necessary and ...

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“…The use of this result in NIEP goes back to Perfect who first applied it in [6], obtaining conditions which are sufficient for the realizability of partitioned real spectra, with the partition allowing some of its pieces to be nonrealizable. This result was somehow abandoned for many years until Soto and Rojo rediscovered it in [7] and since then, it has given rise to a number of sufficient conditions for the existence and construction of nonnegative matrices with prescribed real or complex spectrum (see [6][7][8][9][10] and the references therein):…”

Section: Introductionmentioning

confidence: 99%

Normal nonnegative realization of spectra

Julio

Manzaneda

Soto

2014

Linear and Multilinear Algebra

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The nonnegative inverse eigenvalue problem is the problem of finding necessary and sufficient conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum. This problem remains open for n ≥ 5. If the matrix A is required to be normal, the problem will be called the normal nonnegative inverse eigenvalue problem (NNIEP). Sufficient conditions for a list of complex numbers to be the spectrum of a normal nonnegative matrix were obtained by Xu [Linear Multilinear Algebra. 1993;34:353-364]. In this paper, we give a normal version of a rank-r perturbation result due to Rado and published by Perfect [Duke Math. J. 1955;22:305-311], which allow us to obtain new sufficient conditions for the NNIEP to have a solution. These new conditions significantly improve Xu's conditions. We also apply our results to construct nonnegative matrices with arbitrarily prescribed elementary divisors.

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Nonnegative matrices with prescribed spectrum and elementary divisors

Cited by 26 publications

References 12 publications

A family of nonnegative matrices with prescribed spectrum and elementary divisors

A family of nonnegative matrices with prescribed spectrum and elementary divisors

Nonnegative Inverse Elementary Divisors Problem

Normal nonnegative realization of spectra

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