Unipotent diagonalization of matrices

Authors : Grigore CALUGAREANU

Pages : 71-87

Doi:10.24330/ieja.1281654

View : 21 | Download : 85

Publication Date : 2023-07-10

Article Type : Research Paper

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