Sign patterns that allow minimal semipositivity Original Research Article

A real m-by-n matrix A is semipositive if there is a vector x greater-or-equal, slanted 0 such that Ax> 0, the inequalities being entrywise. A is minimally semipositive if A is semipositive and no column-deleted submatrix of A is semipositive. We characterize the sign patterns of (possibly nonsquare) minimally semipositive matrices which have no zero entries. The work depends strongly on previous results by Johnson, Leighton, and Robinson on the sign patterns of square inverse positive matrices, and a major tool is a theorem concerning complete bipartite subgraphs of bipartite graphs that may have independent interest.