On the matrix equation XA-AX=X^p

@article{Burde2004OnTM,
  title={On the matrix equation XA-AX=X^p},
  author={D. Burde},
  journal={arXiv: Rings and Algebras},
  year={2004}
}
  • D. Burde
  • Published 2004
  • Mathematics
  • arXiv: Rings and Algebras
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 $A^mX^{\ell},X^{\ell}A^m$ and $(AX)^{\ell}$ are given. The case $p=2$ is a special case of the algebraic Riccati equation. 
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