Sign patterns that require a positive or nonnegative left inverse

Abstract

Abstract. An m by n sign pattern A is an m by n matrix with entries in {+, −, 0}. The sign pattern A requires a positive (resp. nonnegative) left inverse provided each real matrix with sign pattern A has a left inverse with all entries positive (resp. nonnegative). In this paper, necessary and sufficient conditions are given for a sign pattern to require a positive or nonnegative left inverse. It is also shown that for n ≥ 2, there are no square sign patterns of order n that require a positive (left) inverse, and that an n by n sign pattern requiring a nonnegative (left) inverse is permutationally equivalent to an upper triangular sign pattern with positive main diagonal entries and nonpositive off-diagonal entries.