Unipotent diagonalization of matrices

Year 2023, Volume: 34 Issue: 34, 71 - 87, 10.07.2023

Grigore Calugareanu

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

Abstract

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 $E_{11}$.

Keywords

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

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