Corpus ID: 119305040

# The matrix equations $XA-AX=X^{\alpha}g(X)$ over fields or rings

@article{Bourgeois2014TheME,
title={The matrix equations \$XA-AX=X^\{\alpha\}g(X)\$ over fields or rings},
author={G. Bourgeois},
journal={arXiv: Rings and Algebras},
year={2014}
}
• G. Bourgeois
• Published 2014
• Mathematics
• arXiv: Rings and Algebras
Let $n,\alpha\geq 2$. Let $K$ be an algebraically closed field with characteristic $0$ or greater than $n$. We show that the dimension of the variety of pairs $(A,B)\in {M_n(K)}^2$, with $B$ nilpotent, that satisfy $AB-BA=A^{\alpha}$ or $A^2-2AB+B^2=0$ is $n^2-1$ ; moreover such matrices $(A,B)$ are simultaneously triangularizable. Let $R$ be a reduced ring such that $n!$ is not a zero-divisor and $A$ be a generic matrix over $R$ ; we show that $X=0$ is the sole solution of \$AX-XA=X^{\alpha… Expand

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