Matrix Links, An Extremization Problem, and the Reduction of a Non-Negative Matrix to One With Prescribed Row and Column Sums

Abstract

The word matrix will, in this paper, always connote a matrix with non-negative elements, having in general m rows and n columns. In conformity with the definition in (4), two matrices will be said to have the same pattern if the entry in any row or column is zero or not according as the corresponding entry of the other is zero or not. The symbols ρi and σj will stand for the respective phrases “ith row-sum” and “jth column-sum” of the matrix under consideration. r1 … , rm; c1, … , cn is a set of positive numbers. It will be said to be consistent for the pattern of an m × n matrix, if there exists a matrix of that pattern for which ρi = ri and σj = cj for all i and j.