The number of standard forms of matrices over imaginary Euclidean quadratic rings with respect to the $(z,k)$–equivalence

N. B. Ladzoryshyn; V. M. Petrychkovych

doi:10.30970/ms.57.2.115-121

N. B. Ladzoryshyn Pidstryhach Institute for Applied Problems of Mechanics and Mathematics NAS of Ukraine Lviv, Ukraine
V. M. Petrychkovych Pidstryhach Institute for Applied Problems of Mechanics and Mathematics NAS of Ukraine

DOI: https://doi.org/10.30970/ms.57.2.115-121

Keywords: quadratic ring; equivalence of a matrix; (z, k)–equivalence; standard form

Abstract

The $(z,k)$--equivalence of matrices over imaginary Euclidean
quadratic rings is investigated. The classes of matrices over
these rings are selected for which the standard form with respect
to $(z,k)$--equivalence is uniquely defined and equal to the Smith
normal form. It is established that the number of standard forms
over imaginary Euclidean quadratic rings is finite. Bounds for a
number of standard forms are established.

Author Biography

N. B. Ladzoryshyn, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics NAS of Ukraine Lviv, Ukraine

Pidstryhach Institute for Applied Problems of Mechanics

and Mathematics NAS of Ukraine

Lviv, Ukraine

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