A note on the matrix equation XA – AX = Xp Original Research Article

Another, more elementary proof is given of Proposition 2.3 in a recent paper by Dietrich Burde. This proposition says that n × n matrices A and X over an algebraically closed field K of characteristic zero satisfying the equality in the title of this paper can be simultaneously triangularized. The proposed proof is based on the Shemesh criterion for two matrices to have a common eigenvector.