An extension of a theorem of Darroch and Ratcliff in loglinear models and its application to scaling multidimensional matrices

Let C=(c<SUB>ij</SUB>) be an m ×n matrix with real entries. Let b be any nonzero m-vector. Let K = {ΠCΠ = b, Π ≥ 0} be bounded. Let x = (x<SUB>1</SUB>, x<SUB>2</SUB>,...,x<SUB>n</SUB>), y = (y<SUB>1</SUB>, y<SUB>2</SUB>,...,y<SUB>n</SUB>) be two nonnegative vectors with y ∈ K and x<SUB>j</SUB> = 0 ⇔ y<SUB>j</SUB> = 0 for any coordinate j. Then it is shown that there exists a Π ∈ K and positive numbers z<SUB>1</SUB>, z<SUB>2</SUB>,..., z<SUB>m</SUB> such that π<SUB>j</SUB> = xj∈mi = 1 zciji for all j. Th theorem slightly generalizes a theorem of Darroch and Ratcliff in loglinear models with a completely different proof technique. The proof relies on an extension of a topological theorem of Kronecker to set valued maps and the duality theorem of linear programming. Many theorems in scaling of matrices and multidimensional matrices are direct consequences of this theorem. The main idea is to associate a suitable zero-one matrix of transportation with any multidimensional matrix. Some motivations for scaling applications are also discussed