On entropy-preserving stochastic averages (Q627945)

The stochastic average of an \(n\) by \(n\) doubly stochastic matrix \(A\) and an \(n\)-long probability distribution \(P\) is defined to be the probability distribution \(AP\). This paper studies the set \(\Gamma_n\) of ordered pairs \((A,P)\) whose stochastic averages preserve entropy, i.e. \(H(AP)=H(P)\). Several algebraic characterizations of \(\Gamma_n\) are derived. Using these, the geometry, topology and combinatorial structure of \(\Gamma_n\) and certain of its distinguished subsets are elucidated. For example, \((A,P)\) is in \(\Gamma_n\) if and only if \(A^tAP=P\). It is shown that \(\Gamma_n\) is a PL-contractible subset of the appropriate Euclidean space. Readers studying not only linear algebra but also graph theory and combinatorial mathematics (algorithms, information theory, topology) may have interest in this article. Numerous examples are given to support the reader's understanding.