About the spectra of a real nonnegative matrix and its signings

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].