Universal realizability in low dimension
This paper deals with the nonnegative inverse eigenvalue problem (NIEP) for matrices of size \(n \times n\), \(n \leq 5\). The aim is to find necessary and sufficient conditions for a list \(\Lambda=\{\lambda_1,\dots,\lambda_n\}\) of complex numbers to be the spectrum of a nonnegative matrix \(A\). \(\Lambda\) is said to be realizable and \(A\) is a realizing matrix. A realizable list \(\Lambda\) is called universally realizable (UR) if, for every possible Jordan canonical form allowed by \(\Lambda\), there is a nonnegative matrix with spectrum \(\Lambda\). The problem of finding necessary and sufficient conditions for a realizable list \(\Lambda\) to be universally realizable is called universal realizability problem (URP). It is clear that for \(n \leq 3\) the concepts of universally realizable and realizable are equivalent. The case \(n=4\) is easily deduced from previous results of the authors. The URP has been completely solved, with different approaches, for different types of lists. Here, the authors characterize the universal realizability of real spectra of size 5 and trace zero by applying several criteria and results from the NIEP. In addition, they describe a region for the universal realizability of non-real 5-spectra with trace zero. The authors distinguish two cases: (a) non-real spectra with repeated complex eigenvalues, and (b) non-real spectra with repeated real eigenvalues. To prove the nonexistence of certain Jordan canonical forms, the authors use methods based on graph theory. Finally, the authors show that realizable lists on the left half-plane, that is, lists \(\Lambda=\{\lambda_1,\dots,\lambda_n\}\), where \(\lambda_1\) is the Perron eigenvalue and \(\mathrm{Re} (\lambda_i) \leq 0\), for \(i=1,2,\ldots,n\), are not necessarily universally realizable.
scientific article; zbMATH DE number 7337640
scientific article