Sign Patterns That Allow a Positive or Nonnegative Left Inverse
In‐Jae Kim, Dale D. Olesky, Bryan L. Shader, Pauline Van den Driessche · SIAM Journal on Matrix Analysis and Applications · 2007
An m by n sign pattern $\mathcal{S}$ 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 $\mathcal{S}$ 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 $m>n \geq 2$, 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 $m \geq n \geq 2$ that allow a positive left inverse, giving a generalization of the known result for the square case, which involves a related bipartite digraph. For $m \geq n$, m by n sign patterns with all entries in $\{+,0\}$ and m by 2 sign patterns with $m \geq 2$ 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.