Doubly stochastic and permutation solutions to AXA = XAX when A is a permutation matrix

Abstract

Let A be a permutation matrix. We prove that the equation has nontrivial doubly stochastic solutions if and only if A has at least one nonzero entry on its main diagonal. We provide an algorithm for obtaining some of these doubly stochastic solutions. Afterwards, we completely characterize the set of permutation solutions for this equation: we obtain the if and only if criteria for their existence, provide an algorithm for the exact solutions in closed form, prove that they are all non-commuting solutions for the initial equation and we calculate the cardinality of this set. These findings generalize some previously known results regarding this topic.