How many negative entries can A(2) have?

A matrix whose entries are +, -, and 0 is called a sign pattern matrix. We first characterize sign patterns A such that A(2) less than or equal to 0. Further, we determine the maximum number of negative entries that can occur in A(2) whenever A(2) less than or equal to 0, and then we characterize the sign patterns that achieve this maximum number. Next we find the maximum number of negative entries that can occur in the square of any sign pattern matrix, and provide a class of sign patterns that achieve this maximum. We also determine the maximum number of negative entries in the square of any real matrix. Finally, we discuss the spectral properties of the sign patterns whose squares contain the maximum number of negative entries in the special case when A(2) less than or equal to 0, and in the general case that includes any sign pattern. (C) Elsevier Science Inc., 1997.