Spectra inhabiting the left half-plane that are universally realizable

Ricardo L Soto

doi:10.1515/spma-2021-0155

Abstract

Abstract Let Λ = {λ1, λ2, . . ., λ n } be a list of complex numbers. Λ is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. Minc ([21],1981) showed that if Λ is diagonalizably positively realizable, then Λ is universally realizable. The positivity condition is essential for the proof of Minc, and the question whether the result holds for nonnegative realizations has been open for almost forty years. Recently, two extensions of the Minc’s result have been proved in ([5], 2018) and ([12], 2020). In this work we characterize new left half-plane lists (λ1 > 0, Re λ i ≤ 0, i = 2, . . ., n) no positively realizable, which are universally realizable. We also show new criteria which allow to decide about the universal realizability of more general lists, extending in this way some previous results.

Highlights

A list Λ = {λ, λ, . . . , λn} of complex numbers is said to be realizable, if it is the spectrum of an n-by-n nonnegative matrix A
We show new criteria which allow to decide about the universal realizability of more general lists, extending in this way some previous results
Since that any list of complex numbers is the spectrum of some matrix, we will sometimes use the word spectrum instead of the word list

Summary

Introduction

In [5], the authors prove that if a list Λ of complex numbers is the spectrum of a nonnegative diagonalizable matrix A ∈ CSλ with a positive row or column, Λ is UR. In [12], the authors prove that if a list Λ of complex numbers is diagonalizably ODP realizable, that is, if Λ is the spectrum of a diagonalizable nonnegative matrix with only o -diagonal positive entries (zero entries are allowed on the diagonal), Λ is UR. Observe that from Theorem 2.1, to decide about the universal realizability of a list Λ of complex numbers, we do not need to compute a nonnegative matrix with spectrum Λ for each JCF allowed by Λ. Observe that from theorems to decide on the universal realizability of a conjugate-even list Λ of complex numbers, it is enough to show that Λ is realizable by a ODP circulant matrix

Spectra inhabiting the left half plane that are universally realizable III

Then the matrices

More general spectra that are universally realizable