About the spectra of a real nonnegative matrix and its signings

TL;DR

This paper investigates the spectra of signings of a nonnegative real matrix, characterizing when the spectrum of a signing is a scalar multiple of the original spectrum by a complex unit, generalizing previous results.

Contribution

It introduces a comprehensive study of signings of nonnegative matrices with spectra scaled by complex units, extending prior research in the field.

Findings

Characterization of signings with spectra scaled by complex units

Generalization of previous spectral results for nonnegative matrices

Conditions under which signings preserve spectral properties

Abstract

For a real matrix $M$, we denote by $sp(M)$ the spectrum of $M$ and by $\left \vert M\right \vert $ its absolute value, that is the matrix obtained from $M$ by replacing each entry of $M$ by its absolute value. Let $A$ be a nonnegative real matrix, we call a \emph{signing} of $A$ every real matrix $B$ such that $\left \vert B\right \vert =A$. In this paper, we study the set of all signings of $A$ such that $sp(B)=\alpha sp(A)$ where $\alpha$ is a complex unit number. Our work generalizes some results obtained in [1, 5, 8].