A conforming decomposition theorem, a piecewise linear theorem of the alternative, and scalings of matrices satisfying lower and upper bounds

A scaling of a nonnegative matrix A is a matrix XA Y -I. where X and Yare nonsingular. nonnegative diagonal matrices. Some condition may be imposed on the scaling, for exa'mple. when A is square, X = Y or det X = det Y. 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.