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∕n, 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.
