On new universal realizability criteria (Q2105351)

A list \(\Lambda = \{\lambda_1, \lambda_2, \dots, \lambda_n\}\) (a list is used instead of a set to handle multiplicities) of complex numbers is said to be \textit{realizable} if it is the spectrum (counting multiplicities) of a nonnegative matrix and is said to be universally realizable (UR), if it is realizable for each possible Jordan canonical form allowed by \(\Lambda\). In 1981, \textit{H. Minc} [Proc. Am. Math. Soc. 83, 665--669 (1981; Zbl 0472.15006)] proved that if \(\Lambda\) is diagonalizably positively realizable, then \(\Lambda\) is UR. The question whether this result holds for nonnegative realizations was open for almost 40 years. Two extensions of Mins's result have been obtained by \textit{M. Collao} et al. [Spec. Matrices 6, 301--309 (2018; Zbl 1404.15027)] and \textit{C. R. Johnson} et al. [Linear Algebra Appl. 587, 302--313 (2020; Zbl 1475.15012)]. In this paper, the authors exploit these extensions to generate new universal realizability criteria. Moreover, they also prove that under certain conditions, the union of two UR lists is also UR, and for certain criteria, if \(\Lambda\) is UR, then for \(t\ge 0\), \(\Lambda_t = \{\lambda_1 +t, \lambda_2 \pm t, \lambda_3, \dots, \lambda_n\}\) is also UR.