On the comparison of some realizability criteria for the real nonnegative inverse eigenvalue problem

A result by Brauer shows how to modify one single eigenvalue of a matrix via a rank-one perturbation, without changing any of the remaining eigenvalues. This, together with the properties of real matrices with constant row sums, was exploited by the authors in a previous work in connection with the nonnegative inverse eigenvalue problem, obtaining conditions which are sufficient for the existence of an entrywise nonnegative matrix with prescribed spectrum. In this work we make use of Brauer's Theorem again, to show that most of the previous results giving sufficient conditions for the real nonnegative inverse eigenvalue problem can be derived by using Brauer's Theorem. Moreover, the technique is constructive, and there is an algorithmic procedure to construct a matrix realizing the spectrum. In particular, we show that if either Kellogg's realizability criterion or Borobia's realizability criterion is satisfied, then Soto's realizability criterion is also satisfied. None of the converses are true. Thus, the condition given by Soto appears to be the most general sufficient condition so far for the real nonnegative inverse eigenvalue problem.