On the realizability of the critical points of a realizable list

TL;DR

This paper proves Johnson's and Monov's conjectures for various classes of spectra in the nonnegative inverse eigenvalue problem, advancing understanding of the critical points of realizable spectra.

Contribution

It establishes Johnson's and Monov's conjectures for important classes of spectra, including Ciarlet, Sulee7manova, and spectra from specific matrix constructions.

Findings

Johnson's conjecture proven for Ciarlet and Sulee7manova spectra

Monov's conjecture confirmed for spectra realizable via specific matrices

New results on differentiators, trace vectors, and circulant matrices

Abstract

The nonnegative inverse eigenvalue problem (NIEP) is to characterize the spectra of entrywise nonnegative matrices. A finite multiset of complex numbers is called realizable if it is the spectrum of an entrywise nonnegative matrix. Monov conjectured that the k\textsuperscript{th}-moments of the list of critical points of a realizable list are nonnegative. Johnson further conjectured that the list of critical points must be realizable. In this work, Johnson's conjecture, and consequently Monov's conjecture, is established for a variety of important cases including Ciarlet spectra, Sule\u{\i}manova spectra, spectra realizable via companion matrices, and spectra realizable via similarity by a complex Hadamard matrix. Additionally we prove a result on differentiators and trace vectors, and use it to provide an alternate proof of a result due to Malamud and a generalization of a result due to Kushel and Tyaglov on circulant matrices. Implications for further research are discussed.