The nonnegative inverse eigenvalue problem (NIEP) is the problem of characterizing all possible spectra of entrywise nonnegative matrices. It is said that a list \(\Lambda = \{\lambda_1, \dots, \lambda_n\}\) of complex numbers is realizable if it is the spectrum of a nonnegative matrix \(A\). In such a case \(A\) is called realizing matrix.
This survey paper does not contain new results and its goal is to show and emphasize the relevance and importance of some results by A. Brauer [Duke Math. J. 19, 75–91 (1952; Zbl 0046.01202)] in the study of NIEP.
Brauer’s theorem shows how to modify one single eigenvalue of a matrix, via a rank-one perturbation, without changing any of the remaining eigenvalues. Rado’s theorem is an extension of Brauer’s theorem and it shows how to modify \(r\) eigenvalues of a \(n\times n\) matrix with \(r < n\) via a rank \(r\) perturbation, without changing any of the remaining \(n-r\) eigenvalues. Rado’s theorem was introduced and applied by H. Perfect [Duke Math. J. 22, 305–311 (1955; Zbl 0068.32704)]. Virtually all known results, which give sufficient conditions for \(\Lambda\) to be realizable or universally realizable, are obtainable from results by Brauer in a constructive way.