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Sign Patterns That Allow a Positive or Nonnegative Left Inverse

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 â‰¥ 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 â‰¥ n â‰¥ 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 â‰¥ 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

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##### Keywords

bipartite digraph , nonnegative left inverse , positive left inverse , positive left null-vector , sign pattern , strong Hall , Mathematics