On the nonnegative eigenvalue problem

On Open Problems of Nonnegative Inverse Eigenvalues Problem

“…R C Problem 1, Problem 2 and Problem 3 have not been solved yet since they were presented. To find the solvability of these three problems, people have been studying them in the recent 70 years (refer to the references), the achievements people have got and their limitations and practical application depict were evaluated in scholarly treatise [7] (Moody T. Chu, Gene H. Golub, Inverse Eigenvalue Problems: Theory, Algorithms, and Applications, Oxford University press. P 93-122) and article [8] (Ricardo L.SoTo, Reliability by symmetric nonnegative matrces, http://www.scielo.cl/pdf/proy/v24n1/art06.…”

“…Readers also may refer to [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] for some previous results. In some articles, some necessary conditions and sufficient conditions for the three problems above have been given under some small dimension or special cases [7]. Also See the survey paper [26] and the book [2, Chapter VII].…”

Symmetric nonnegative realization of spectra

“…Kellogg's criterion [8], for instance, was shown by Fiedler [4] to imply symmetric realizability. Radwan [16] proved that Borobia's criterion [1] is also a criterion for symmetric realizability , and Soto's criterion for the RNIEP [19] (Theorem 4.2 below), which contains both Kellogg's and Borobia's criteria [20], was shown in [23] to be also a symmetric realizability criterion. In this section we compare the new result in this paper (Theorem 3.1) with some previous realizability criteria for SNIEP.…”

“…The number −λ n − Sj <0 S j is the minimum value that λ 1 may take in order that Λ be symmetrically realizable according to Theorem 4.1. Now, suppose that Λ = {λ 1 …”

“…The complex cases n = 4 and n = 5 have been solved for matrices of trace zero by Reams [17] and Laffey and Meehan [10], respectively. Sufficient conditions or realizability criteria for the existence of a nonnegative matrix with a given real spectrum have been obtained in [25,14,15,18,8,1,19,22] (see [3, §2.1] and references therein for a comprehensive survey). If we additionally require the realizing matrix to be symmetric, we have the symmetric nonnegative inverse eigenvalue problem (hereafter SNIEP).…”