About the spectra of a real nonnegative matrix and its signings

Abstract

For a complex matrix \(M\), we denote by \(\operatorname{Sp}(M)\) the spectrum of \(M\) and by \(|M|\) 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 signing of \(A\) every real matrix \(B\) such that \(|B| =A\). In this paper, we characterize the set of all signings of \(A\) such that \(\operatorname{Sp}(B)=\alpha \operatorname{Sp}(A)\) where \(\alpha\) is a complex unit number. Our motivation comes from some recent results about the relationship between the spectrum of a graph and the skew spectra of its orientations.