Construction and nonnegativity of sign idempotent sign pattern matrices

In this paper, we modify Eschenbach’s algorithm for constructing sign idempotent sign pattern matrices so that it correctly constructs all of them. We find distinct classes of sign idempotent sign pattern matrices that are signature similar to an entrywise nonnegative sign pattern matrix. Additionally, if for a sign idempotent sign pattern matrix A there exists a signature matrix S such that SAS is nonnegative, we prove such S is unique up to multiplication by -1 if the signed digraph D(A) is not disconnected.