Sign patterns that require a positive or nonnegative left inverse
نویسندگان
چکیده
An m by n sign pattern A is an m by n matrix with entries in {+,−, 0}. The sign pattern A requires a positive (resp. nonnegative) left inverse provided each real matrix with sign pattern A has a left inverse with all entries positive (resp. nonnegative). In this paper, necessary and sufficient conditions are given for a sign pattern to require a positive or nonnegative left inverse. It is also shown that for n ≥ 2, there are no square sign patterns of order n that require a positive (left) inverse, and that an n by n sign pattern requiring a nonnegative (left) inverse is permutationally equivalent to an upper triangular sign pattern with positive main diagonal entries and nonpositive off-diagonal entries.
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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 = In, where In is the n by n identity matrix. For m > n ≥ 2, a characterization of m by n sign patt...
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