The manuscript deals with nonnegative inverse eigenvalue problems. The set of complex numbers \(\Lambda=\{\lambda_1, \lambda_2, \ldots, \lambda_n\}\) is said to be realizable, if there is an \(n \times n\) nonnegative matrix \(A\) with spectrum \(\Lambda\), with \(\lambda_1\) being the Perron eigenvalue. In this case, \(A\) is said to be a realizing matrix. \(\Lambda\) is diagonalizably realizable if a realizing matrix for \(\Lambda\) is diagonalizable, and it is universally realizable if it is realizable for every possible Jordan canonical form allowed by \(\Lambda\).
In this paper, the authors present new realizability criteria for list of complex numbers, extending the criteria usually called Soto1 and Soto2. These criteria are sufficient conditions for the real nonnegative inverse eigenvalue problem. On the other hand, a diagonalizable version of a perturbation result by R. Rado is also proved (see [H. Perfect, Duke Math. J. 22, 305–311 (1955; Zbl 0068.32704)]). It is useful for constructing diagonalizable nonnegative matrices, with prescribed spectrum, and for deciding about the universal realizability of spectra.
Finally, the authors establish an universal realizability criterion for a list of complex numbers \(\Lambda=\{\lambda_1, \lambda_2, \ldots, \lambda_n\}\).