Symmetric nonnegative realization of spectra

Abstract

Abstract.A perturbation result, due to R. Rado and presented by H. Perfect in 1955, shows how to modify r eigenvalues of a matrix of order n, r ≤ n, via a perturbation of rank r, without changing any of the n − r remaining eigenvalues. This result extended a previous one, due to Brauer, on perturbations of rank r = 1. Both results have been exploited in connection with the nonnegative inverse eigenvalue problem. In this paper a symmetric version of Rado's extension is given, which allows us to obtain a new, more general, sufficient condition for the existence of symmetric nonnegative matrices with prescribed spectrum.Key words. Symmetric nonnegative inverse eigenvalue problem.

Nonnegative realization of complex spectra

“…Since we are interested in the nonnegative realization of spectra, in this section, we show how to construct an n × n Hermitian matrix with prescribed spectrum, whose entries have nonnegative real part. In particular, we generalize a result in [10] for symmetric nonnegative matrices. The technique is the same as in Section 2.…”

On nonnegative realization of partitioned spectra

“…In [8] Laffey andŠmigoc introduce a symmetric realizability criterion, which combine realizability and diagonal elements of realizing matrices. In [19], the authors give a symmetric version of the Rado's result and, from it, obtain a symmetric realizability criterion, Theorem 1.4 in this paper, which contains criteria of Kellogg, Borobia and Soto. In this Section we extend Theorem 1.4.…”

“…In [19], using the Fiedler's conditions, the authors show how to construct such a matrix B for r ≥ 3. Now, from Theorem 4.1, we have the following sufficient condition.…”

“…In this paper, we shall extend this sufficient condition. In [19], the authors prove a symmetric version of the Rado's result, and from it, they obtain a new symmetric realizability criterion for partitioned lists, which will also be extended in this paper. Recently, in [20], the authors have derived an efficient and constructive sufficient condition for the existence of a nonnegative matrix with a prescribed complex spectrum.…”

“…In [10] the authors construct a map of the sufficient conditions for the RNIEP and establish inclusion relations or independency relations between different sufficient conditions. If the realizing matrix is required to be symmetric, then we have the symmetric nonnegative inverse eigenvalue problem (SNIEP) [5,8,13,17,19,21].…”