About the spectra of a real nonnegative matrix and its signings

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.