REDUCTION OF A MATRIX WITH POSITIVE ELEMENTS TO A DOUBLY STOCHASTIC MATRIX

expressed in the form T = D1A D2, where D1 and D2 are diagonal matrices with strictly positive diagonal elements. The matrices D1 and D2 are themselves unique up to a scalar factor. The existence of T, D1 and D2 is established by a "constructive" but "limiting" procedure in [2]. There the author also shows by means of specific examples that the assumption made in the theorem that the elements of A are strictly positive cannot, in general, be relaxed to the one that they are nonnegative. In this paper, we give an "existence proof" of the theorem whose main advantage besides its simplicity is that it not only makes clear why the theorem holds, but also explains the reasons why the assumption that the elements be positive cannot, in general, be replaced by the assumption that they be only nonnegative. The proof, in outline, is as follows. It is shown that A defines in a natural manner a continuous mapping of a certain subset of a finite dimensional Euclidean space into itself, which maps lines through the origin onto lines. By an appeal to a fixed-point theorem, one finds that the mapping leaves a line fixed. This means that the co-ordinates of any point on the line are proportional to those of its image on the line. A simple substitution establishes next that the constant of proportionality is unity. Thus all points on the line are fixed points. The co-ordinates of any one of those points form the diagonal elements of the diagonal matrix D2 of the theorem, and the other diagonal matrix D1 is determined at once by the mapping. 2. Proof. Solely in order to decrease the complexity of notation, it will be assumed that A is a 3 X3 matrix with positive entries, Ia b c'