On entropy-preserving stochastic averages

When an n×n doubly stochastic matrix A acts on Rⁿ on the left as a linear transformation and P is an n-long probability vector, we refer to the new probability vector AP as the stochastic average of the pair (A,P). Let Γ_n denote the set of pairs (A,P) whose stochastic average preserves the entropy of P:H(AP)=H(P). Γ_n is a subset of B_n×Σ_n where B_n is the Birkhoff polytope and Σ_n is the probability simplex.We characterize Γ_n and determine its geometry, topology,and combinatorial structure. For example, we find that (A,P)∈Γ_n if and only if A^tAP=P. We show that for any n, Γ_n is a connected set, and is in fact piecewise-linearly contractible in B_n×Σ_n. We write Γ_n as a finite union of product subspaces, in two distinct ways. We derive the geometry of the fibers (A,·) and (·,P) of Γ_n. Γ₃ is worked out in detail. Our analysis exploits the convexity of xlogx and the structure of an efficiently computable bipartite graph that we associate to each ds-matrix. This graph also lets us represent such a matrix in a permutation-equivalent, block diagonal form where each block is doubly stochastic and fully indecomposable.