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 eq ual 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 the n 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 pat terns that achieve this maximum. We also determine the maximum number of negative entries in the square of any real matrix. Finally, we disc uss 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 sig n pattern. (C) Elsevier Science Inc., 1997.