On the inverse eigenvalue problem of symmetric nonnegative matrices (original) (raw)

A note on the symmetric nonnegative inverse eigenvalue problem

The symmetric nonnegative inverse eigenvalue problem is the problem of characterizing all possible spectra of n × n symmetric entrywise nonnegative matrices. The problem remains open for n ≥ 5. A number of realizability criteria or sufficient conditions for the problem to have a solution are known. In this paper we show that most of these sufficient conditions can be obtained by the use of a result by Soto, Rojo, Moro, Borobia in [ELA 16 (2007) 1-18]. Moreover, by applying this result we may always compute a solution matrix.

A map of sufficient conditions for the symmetric nonnegative inverse eigenvalue problem

Linear Algebra and its Applications

The real nonnegative inverse eigenvalue problem (RNIEP) is the problem of determining necessary and sufficient conditions for a list of real numbers to be the spectrum of an entrywise nonnegative matrix. A number of sufficient conditions for the existence of such a matrix are known. In this paper, in order to construct a map of sufficient conditions, we compare these conditions and establish inclusion relations or independency relations between them.

On the nonnegative inverse eigenvalue problem of traditional matrices

In this paper, at �rst for a given set of real or complex numbers � with nonnegative summation, we introduce some special conditions that with them there is no nonnegative tridiagonal matrix in which � is its spectrum. In continue we present some conditions for existence such nonnegative tridiagonal matrices.

Realizability criterion for the symmetric nonnegative inverse eigenvalue problem

Linear Algebra and its Applications, 2006

Let = {λ 1 , λ 2 ,. .. , λ n } a set of real numbers. The real nonnegative inverse eigenvalue problem (RNIEP) is the problem of determining necessary and sufficient conditions in order that be the spectrum of an entrywise nonnegative n×n matrix. If there exists a nonnegative matrix A with spectrum we say that is realized by A. Many realizability criteria for the existence of such a matrix A are known. This paper shows that a realizability criterion given by the author, which contains both Kellogg's realizability criterion and Borobia's realizability criterion, is sufficient for the existence of an n×n symmetric nonnegative matrix with prescribed spectrum .

The Diagonalizable Nonnegative Inverse Eigenvalue Problem

Special Matrices

In this articlewe provide some lists of real numberswhich can be realized as the spectra of nonnegative diagonalizable matrices but which are not the spectra of nonnegative symmetric matrices. In particular, we examine the classical list σ = (3 + t, 3 − t, −2, −2, −2) with t ≥ 0, and show that 0 is realizable by a nonnegative diagonalizable matrix only for t ≥ 1. We also provide examples of lists which are realizable as the spectra of nonnegative matrices, but not as the spectra of nonnegative diagonalizable matrices by examining the Jordan Normal Form

ON A CLASSIC EXAMPLE IN THE NONNEGATIVE INVERSE EIGENVALUE PROBLEM

2008

This paper presents a construction of nonnegative matrices with nonzero spectrum τ =( 3 +t, 3 − t, −2, −2, −2) for t> 0. The result presented gives a constructive proof of a result of Boyle and Handelman in this special case. This example exhibits a surprisingly fast convergence of the spectral gap of τ to zero as a function

A map of sufficient conditions for the real nonnegative inverse eigenvalue problem

Linear Algebra and its Applications, 2007

The symmetric nonnegative inverse eigenvalue problem for 5×5 matrices

Linear Algebra and its Applications, 2004

The symmetric nonnegative inverse eigenvalue problem (SNIEP) asks when a list σ = (λ 1 , λ 2 , . . . , λ n ) of n real numbers is the spectrum of an n×n symmetric nonnegative matrix. This problem is completely solved only for n ≤ 4. Our main goal here is to contribute to the solution of SNIEP for n = 5. We also give a sufficient condition for a list σ to be realized as the spectrum of a symmetric positive matrix.

THE REAL AND THE SYMMETRIC NONNEGATIVE INVERSE EIGENVALUE PROBLEMS ARE DIFFERENT

2000

We show that there exist real numbers 1; 2;:::;n that occur as the eigenvalues of an entry-wise nonnegative n-by-n matrix but do not occur as the eigenvalues of a symmetric nonnegative n-by-n matrix. This solves a problem posed by Boyle and Handelman, Hershkowitz, and others. In the process, recent work by Boyle and Handelman that solves the nonnegative inverse eigenvalue

On the inverse eigenvalue problem of symmetric nonnegative matrices (original) (raw)

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