Primitive Zero-Symmetric Sign Pattern Matrices with Zero Diagonal Attaining the Maximum Base

A sign pattern matrix or sign pattern A is a matrix whose entries are from the set {1,−1, 0}. Notice that for a square sign pattern matrixA, in the computation of the signs of the entries of the power A, an ambiguous sign may arise when a positive sign is added to a negative sign. So a new symbol # was introduced in 1 to denote such an ambiguous sign. The powers of a square sign pattern have been investigated to some extent, see, for example, 1–12 . In 1 , the set Γ {1,−1, 0, #} is defined as the generalized sign set and the matrices with entries in the set Γ are called generalized sign pattern matrices, and the addition and multiplication involving the symbol # are defined as follows: