Unipotent diagonalization of matrices

Yıl 2023, Cilt: 34 Sayı: 34, 71 - 87, 10.07.2023

Grigore Calugareanu

https://doi.org/10.24330/ieja.1281654

Öz

An element $u$ of a ring $R$ is called \textsl{unipotent} if $u-1$ is
nilpotent. Two elements $a,b\in R$ are called \textsl{unipotent equivalent}
if there exist unipotents $p,q\in R$ such that $b=q^{-1}ap$. Two square
matrices $A,B$ are called \textsl{strongly unipotent equivalent} if there
are unipotent triangular matrices $P,Q$ with $B=Q^{-1}AP$.
In this paper, over commutative reduced rings, we characterize the matrices
which are strongly unipotent equivalent to diagonal matrices. For $2\times 2$
matrices over B\'{e}zout domains, we characterize the nilpotent matrices
unipotent equivalent to some multiples of $E_{12}$ and the nontrivial
idempotents unipotent equivalent to $E_{11}$.

Anahtar Kelimeler

Unipotent equivalent, strongly unipotent equivalent matrix, ue- diagonalizable matrix, nilpotent matrix, idempotent matrix

Kaynakça

• G. Almkvist, Endomorphisms of finitely generated projective modules over a commutative ring, Ark. Mat., 11 (1973), 263-301.
• A. J. Diesl, Nil clean rings, J. Algebra, 383 (2013), 197-211.
• K. Matthews, Solving the Diophantine equation $ax^{2}+bxy+cy^{2}+dx+ey+f=0$, preprint, 2015-2020.
• K. Matthews, http://www.numbertheory.org/php/generalquadratic.html.
• J. Ster, Rings in which nilpotents form a subring, Carpathian J. Math., 31(2) (2015), 157-163.