Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem

A result by Brauer, which shows how to modify one single eigenvalue of a matrix without changing any of the remaining eigenvalues, plays a relevant role in the study of the nonnegative inverse eigenvalue problem (NIEP). Perfect, in a long time ignored paper (1955) presents an extension of this result, which shows how to modify r eigenvalues of a matrix of order n, r < n, via a rank-r perturbation, without changing any of the n − r remaining eigenvalues. By using this extension, Perfect gives a realizability criterion for the real NIEP, which is not contained in Soto’s realizability criterion. In this work, by extending Perfect’s result, we give a new realizability criterion for the real NIEP, which contains Soto’s criterion. Thus, this new realizability criterion appears to be the most general sufficient condition for the real NIEP so far. We also contribute to the solution of the symmetric NIEP for n = 5.