Sign idempotent sign patterns similar to nonnegative sign patterns (Q924325)

An \(n\times n\) matrix \(A\) \(=[a_{ij}] \) whose entries are from the set \(\{+,-,0\}\) is called a sign pattern. Using the natural product in \(\{ +,-,0\}\), the power \(A^{2}\) is defined as a sign pattern provided, for each \(i,j\), no two products \(a_{ik}a_{kj}\) (\(k=1,\dots,n\)) have opposite signs. If \(A^{2}\) is a sign pattern and \(A=A^{2}\) then \(A\) is called sign idempotent. The author points out that two of the main theorems of \textit{C. Eschenbach} [Linear Algebra Appl. 180, 153--165 (1993; Zbl 0777.05032)] characterizing sign idempotent matrices are false, and gives counterexamples. He then presents two classes of sign idempotents which are similar to nonnegative sign patterns and answers and gives some general criteria for when there exists a real \(n\times n\) idempotent matrix \(B\) such that \(\text{sign}(B)=A\).