Realizability criterion for the symmetric nonnegative inverse eigenvalue problem

Let Λ = {λ1, λ2, … , λn} a set of real numbers. The real nonnegative inverse eigenvalue problem (RNIEP) is the problem of determining necessary and sufficient conditions in order that Λ be the spectrum of an entrywise nonnegative n × n matrix. If there exists a nonnegative matrix A with spectrum Λ we say that Λ 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 Λ.