A note on the matrix equation \(XA - AX = X^{p}\) (Q886142)

The author proposes a more elementary proof of Proposition 2.3 in the paper \textit{D. Burde} [Linear Algebra Appl. 404, 147--165 (2005; Zbl 1079.15015)] which says that if for the \(n\times n\)-matrices \(A\), \(X\) over an algebraically closed field one has \(XA-AX=X^p\), \(p\in {\mathbb N}\), \(1<p<n\), then they are simultaneously diagonalizable. The proof proposed in the present paper is based on the Shemesh criterion for two matrices to have a common eigenvector.