Maximum Entropy Convex Decompositions of Doubly Stochastic and Nonnegative Matrices

Two maximum entropy convex decompositions are computed with the use of the iterative proportional fitting procedure. First, a doubly stochastic version of a 5 × 5 British social mobility table is represented as the sum of 120 5 × 5 permutation matrices. The most heavily weighted permutations display a bandwidth form, indicative of relatively strong movements within social classes and between neighboring classes. Then the mobility table itself is expressed as the sum of 6 985 5 × 5 transportation matrices—possessing the same row and column sums as the mobility table. A particular block-diagonal structure is evident in the matrices assigned the greatest weight. The methodology can be applied as well to the representation of other nonnegative matrices in terms of their extreme points, and should be extendable to higher-order mathematical structures—for example, operators and functions.