Idempotence of \((1,-1)\)-matrices (Q2716661)

Let \(A\) denote an \(n\)-square sign pattern matrix, and define the \textit{sign pattern class} of \(A\) by \(Q(A) =\{B\in M_n(\mathbb R):\text{sgn }B=A\}\). \(A\) is said to be \textit{sign idempotent} if \(B^2\in Q(A)\) whenever \(B\in Q(A)\). The authors characterize a class of idempotent \((1,-1)\)-matrices that are not sign idempotent thus giving a partial answer to a problem posed by \textit{C. Eschenbach} [Linear Algebra Appl. 180, 153--165 (1993; Zbl 0777.05032)]. This paper is a continuation of the work by \textit{S.-G. Lee} and \textit{S.-W. Park} [Commun. Korean. Math. Soc. 10, No. 3, 561--573 (1995; Zbl 0945.15018)].