On the scaling of multidimensional matrices (Q1121341)

The authors give a new proof of a result of \textit{R. Bapat} and \textit{T. E. S. Raghavan} [ibid. 114/115, 705-715 (1989; Zbl 0661.15005)]: let \(X=(x_{ijk})\), \(Y=(y_{ijk})\) be \(r\times s\times t\) matrices (tensors) with the same zero pattern; if the one-dimensional marginals \(\sum_{j,k}y_{ijk}\), \(\sum_{i,k}y_{ijk}\), \(\sum_{i,j}y_{ijk}\) are all positive, then there exist positive scalars \(u_ 1,...,u_ r\), \(v_ 1,...,v_ s\), \(w_ 1,...,w_ t\) such that \(\Pi =(x_{ijk}u_ iv_ jw_ k)\) has the same one-dimensional marginals as Y. As a corollary they derive a result of the reviewer [Ann. Math. Stat. 35, 876- 879 (1964; Zbl 0134.253)]: to a given strictly positive \(N\times N\) matrix A there corresponds exactly one doubly stochastic matrix \(T_ A\) of the form \(D_ 1AD_ 2\) where \(D_ 1\) and \(D_ 2\) are diagonal matrices with positive main diagonals; the matrices \(D_ 1\) and \(D_ 2\) are themselves unique up to a scalar factor. The authors also derive other results of the reviewer [Am. Math. Mon. 74, 402-405 (1967; Zbl 0166.037)]: the matrix \(T_ A\) can be obtained as a limit to the iteration of alternately normalizing the row and column sums of A; the rate of convergence is geometric. An estimate of the ratio of convergence is given from which error estimates are then developed.