The matrix equation axb = x

Abstract

This paper considers the solutions of the matrix equation AXB = X where we specify A and B to be n-square and doubly stochastic. Solutions are found explicitly and do not depend on either the Jordan or Rational canonical forms. We further find all doubly stochastic solutions of this equation, by noting that Jn = (1/n), the n-square doubly stochastic matrix in which each entry is 1 In, is always a solution and that the doubly stochastic solutions form a compact convex set. We solve the equation by characterizing the vertices of this convex set. © 1971 Pacific Journal Of Mathematics.