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

References

SHOWING 1-10 OF 11 REFERENCES
On the matrix equation XA-AX=X^p
  • 19
  • PDF
How to solve the matrix equation XA-AX=f(X)
  • 15
  • PDF
Commutators which commute with one factor
  • 13
  • PDF
On the characteristic roots of matric polynomials
  • 68
  • PDF
PAIRS OF MATRICES WITH PROPERTY L. II(
  • 167
  • PDF
A Binomial-like Matrix Equation
  • 3
  • Highly Influential
  • PDF
LOCALIZATIONS OF THE KLEINECKE-SHIROKOV THEOREM
  • 4
  • PDF
Basic algebra: groups
  • rings and fields, Springer
  • 2003
Endomorphisms of finite many projective modules over a commutative ring
  • Arkiv fur matematik
  • 1973
...
1
2
...