Sign Patterns That Allow a Positive or Nonnegative Left Inverse

An m by n sign pattern S is an m by n matrix with entries in {+,-, 0}. Such a sign pattern allows a positive (resp., nonnegative) left inverse, provided that there exist an m by n matrix A with the sign pattern S and an n by m matrix B with only positive (resp., nonnegative) entries satisfying BA = I-n, where I-n is the n by n identity matrix. For m > n ≥ 2, a characterization of m by n sign patterns with no rows of zeros that allow a positive left inverse is given. This leads to a characterization of all m by n sign patterns with m ≥ n ≥ 2 that allow a positive left inverse, giving a generalization of the known result for the square case, which involves a related bipartite digraph. For m = n, m by n sign patterns with all entries in {+, 0} and m by 2 sign patterns with m ≥ 2 that allow a nonnegative left inverse are characterized, and some necessary or sufficient conditions for a general m by n sign pattern to allow a nonnegative left inverse are presented.