Six Matrix Adjustment Problems Solved by Some Fundamental Theorems on Biproportion

Abstract

After defining biproportion (or RAS) rigorously, we recall two fundamental theorems: unicity of biproportion (any biproportional algorithm leads to the same solution than biproportion, which turns biproportion into a mathematical tool as indisputable than proportion), ineffectiveness of separability (premultiplying or post multiplying the initial matrix by a diagonal matrix does not change the biproportional solution) and its corollary (it is equivalent to do a separable modification of the initial matrix or to do a proportional change of each biproportional factors). We then apply these theorems to show immediately that: i) no difficulties are encountered when solving the biproportional program, particularly for the question of the exponential; ii) the equivalence between applying biproportion on coefficient matrices and on transaction matrices is obvious; iii) normalizing the initial values of the biproportional factors obviously do not change anything (even if it is not possible to normalize the final value of these factors unless normalization is scalar); iv) the gravity model is equivalent to biproportion; v) biproportion and entropy give the same result; vi) when ineffectiveness of separability do not hold, the results are different as for added information. To the total, these theorems avoid re-demonstrating most properties.