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Sign Patterns That Allow a Positive or Nonnegative Left Inverse
Kim, I. J. Olesky, D. D. Shader, Bryan L. Van den Driessche, P.
Kim, I. J.
Olesky, D. D.
Shader, Bryan L.
Van den Driessche, P.
Abstract
Description
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.
Date
2006-01-01
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Publisher
University of Wyoming. Libraries
Research Projects
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Keywords
bipartite digraph,nonnegative left inverse,positive left inverse,positive left null-vector,sign pattern,strong Hall,Mathematics