Sign patterns that allow diagonalizability

An n×n sign pattern A allows diagonalizability if there exists a real matrix B in the qualitative class Q(A) of A such that B is diagonalizable. The question of characterizing sign patterns that allow diagonalizability is open. In this paper, we obtain some sufficient conditions for a sign pattern allowing diagonalizability. In particular, it is proved that the combinatorially symmetric sign patterns A allow diagonalizability. We give also two counterexamples for Eschenbach–Johnson’s conjecture in [Linear Algebra Appl. 190 (1993), 169, MR# 94i: 15003].