How many negative entries can A2 have? Original Research Article

A matrix whose entries are +, −, and 0 is called a sign pattern matrix. We first characterize sign patterns A such that A2 less-than-or-equals, slant 0. Further, we determine the maximum number of negative entries that can occur in A2 whenever A2 less-than-or-equals, slant 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 A2 less-than-or-equals, slant 0, and in the general case that includes any sign pattern.