Construction of nonnegative symmetric matrices with given spectrum

The Diagonalizable Nonnegative Inverse Eigenvalue Problem

“…La ey and Smigoc [20] proved that ( + t, − t, − , − , − , ) is symmetrically realizable by a × matrix for t ≥ and we conjecture that t = is the best bound for any number of zeros added to the spectrum σ t .…”

On nonnegative realization of partitioned spectra

“…Fiedler [5], Radwan [13] and Soto [17] proved, respectively, that criteria of Kellogg, Borobia and Soto (see [10]) obtained for the RNIEP are also symmetric realizability criteria. 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 [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].…”

On symmetric nonnegative matrices with prescribed spectrum

“…The first results related with the SNIEP were obtained by Fiedler [2] in 1974. Sufficient conditions for the problem to have a solution have been obtained, among other, in chronological order, in [2,11,8,9,5]. In [12], Spector gives a necessary and sufficient condition for the SNIEP, in the case n = 5 with Based in a result of Horn [4] we give, in section 2, a new constructible sufficient condition for the existence of a symmetric nonnegative matrix with prescribed spectrum.…”