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## Equivalence of pairs of matrices with relatively prime determinants over quadratic rings of principal ideals

Authors: Natalija Ladzoryshyn, Vasyl' Petrychkovych

### Abstract

A special equivalence of matrices and their pairs over quadratic rings is investigated. It is established that for the pair of $\, n\times n \,$ matrices $\, A,B \,$ over the quadratic rings of principal ideals $\, \mathbb{Z}[\sqrt{k}],$ where $\, (detA,detB)=1 \,$, there exist inver\-tible matri\-ces ${\, U\in GL(n,\mathbb{Z})\,}$ and $\, V^A,V^B\in GL(n,\mathbb{Z}[\sqrt{k}])$ such that $\, UAV^A=T^A \,$ and ${\, UBV^B=T^B \, }$ are the lower triangular matrices with invariant factors on the main ${\mbox{diagonals.}}$

Pidstryhach Institute for Applied Problems of Mechanics
and Mathematics of the NAS of Ukraine
3b Naukova Str., 79060, L’viv
Ukraine
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