On the matrix equation XA − AX = Xp

We study the matrix equation XA − AX = Xp in Mn(K) for 1 < p < n. It is shown that every matrix solution X is nilpotent and that the generalized eigenspaces of A are X-invariant. For A being a full Jordan block we describe how to compute all matrix solutions. Combinatorial formulas for AmXℓ, XℓAm and (AX)ℓ are given. The case p = 2 is a special case of the algebraic Riccati equation.