About the spectra of a real nonnegative matrix and its signings

Abstract

For a complex matrix M, we denote by 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 Sp(B) = αSp(A) where α 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.