On new universal realizability criteria

Abstract

A list Λ={λ1,λ2,…,λn}\Lambda =\left\{{\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{n}\right\} of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix and is said to be universally realizable (UR), if it is realizable for each possible Jordan canonical form allowed by Λ\Lambda . In 1981, Minc proved that if Λ\Lambda is diagonalizably positively realizable, then Λ\Lambda is UR [Proc. Amer. Math. Society 83 (1981), 665–669]. The question whether this result holds for nonnegative realizations was open for almost 40 years. Recently, two extensions of Mins’s result have been obtained by Soto et al. [Spec. Matrices 6 (2018), 301–309], [Linear Algebra Appl. 587 (2020), 302–313]. In this work, we exploit these extensions to generate new universal realizability criteria. Moreover, we also prove that under certain conditions, the union of two lists UR is also UR, and for certain criteria, if Λ\Lambda is UR, then for t≥0t\ge 0, Λt={λ1+t,λ2±t,λ3,…,λn}{\Lambda }_{t}=\left\{{\lambda }_{1}+t,{\lambda }_{2}\pm t,{\lambda }_{3},\ldots ,{\lambda }_{n}\right\} is also UR.

Keywords

• nonnegative matrix
• diagonalizable realizability
• universal realizability
• jordan structure
• 15a20
• 15a29