On the matrix equation XA-AX=X^p

Abstract

We study the matrix equation $XA-AX=X^p$ in $M_n(K)$ 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 $A^m{X}^{\ell },X^{\ell }{A}^m$ and $(AX{)}^{\ell }$ are given. The case $p=2$ is a special case of the algebraic Riccati equation.

15 pages