How to solve the matrix equation XA-AX=f(X)

Abstract

Let f be an analytic function defined on a complex domain Ω and A ∈ Mn(C). We assume that there exists a unique α satisfying f (α) = 0. When f ′ (α) = 0 and A is non-derogatory, we completely solve the equation XA − AX = f (X). This generalizes Burde's results. When f ′ (α) = 0, we give a method to solve completely the equation XA − AX = f (X): we reduce the problem to solving a sequence of Sylvester equations. Solutions of the equation f (XA − AX) = X are also given in particular cases.