A note on the real nonnegative inverse eigenvalue problem

Abstract

The Real Nonnegative Inverse Eigenvalue Problem (RNIEP) asks when is a list \[ \sigma=(\lambda_1, \lambda_2,\ldots,\lambda_n)\] consisting of real numbers the spectrum of an $n \times n$ nonnegative matrix $A$. In that case, $\sigma$ is said to be realizable and $A$ is a realizing matrix. In a recent paper dealing with RNIEP, P.~Paparella considered cases of realizable spectra where a realizing matrix can be taken to have a special form, more precisely such that the entries of each row are obtained by permuting the entries of the first row. A matrix of this form is called permutative. Paparella raised the question whether any realizable list $\sigma$ can be realized by a permutative matrix or a direct sum of permutative matrices. In this paper, it is shown that in general the answer is no.