A conforming decomposition theorem,a piecewise linear theorem of the alternative,and scalings of matrices satisfying lower and upper bounds |
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Authors: | Manfred v. Golitschek Uriel G. Rothblum Hans Schneider |
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Affiliation: | 1. Institut für Angewandte Mathematik und Statistik, Universit?t Würzburg, 8700, Würzburg, West Germany 2. School of Organization and Management, Yale University, 06520, New Haven, CT, USA 3. Department of Mathematics, University of Wisconsin, 53706, Madison, WI, USA
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Abstract: | A scaling of a nonnegative matrixA is a matrixXAY ?1, whereX andY are nonsingular, nonnegative diagonal matrices. Some condition may be imposed on the scaling, for example, whenA is square,X=Y or detX=detY. We characterize matrices for which there exists a scaling that satisfies predetermined upper and lower bound. Our principal tools are a piecewise linear theorem of the alternative and a theorem decomposing a solution of a system of equations as a sum of minimal support solutions which conform with the given solutions. |
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