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

Наталя Ладзоришин; V. M. Petrychkovych

doi:10.30970/ms.57.2.115-121

Наталя Ладзоришин Інститут прикладних проблем механіки і математики ім. Я.С.Підстригача НАН України
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

Анотація

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.

Біографія автора

Наталя Ладзоришин, Інститут прикладних проблем механіки і математики ім. Я.С.Підстригача НАН України

Pidstryhach Institute for Applied Problems of Mechanics

and Mathematics NAS of Ukraine

Lviv, Ukraine

Посилання

Zabavskii B.V., Romaniv O.M. Rings with elementary reduction of matrices// Ukr. Math. J. – 2000. – V. 52, №12. – P. 1872–1881.

Mal’tsev A.I. Foundations of Linear Algebra. – Nauka, Moscow, 1970. – 400 p. (in Russian)

Kazimirs’kyi P.S. Decomposition of matrix polynomials into factors. – Lviv: Pidstryhach Inst. Appl. Probl. Mech. and Math. of NAS of Ukrainian, 2015 – 282 p. (in Ukrainian)

Petrychkovich V.M. Semiscalar equivalence and the Smith normal form of polynomial matrices// J. Sov. Math. – 1993. – V.66, №1. – P. 2030–2033.

Dias da Silva J.A., Laffey T.J. On simultaneous similarity of matrices and related questions// Linear Algebra Appl. – 1999. – V. 291. – P. 167–184.

Petrychkovych V.M. Generalized equivalence of matrices and their sets and the factorization of matrices over rings. — Lviv: Pidstryhach Inst. Appl. Probl. Mech. and Math. of NAS of Ukrainian, 2015 – 312 p. (in Ukrainian)

Dzhaliuk N.S., Petrychkovych V.M. Solutions of the matrix linear bilateral polynomial equation and their structure// Algebra Discrete Math. – 2019. – V. 27,№2. – P. 243–251.

Sidorov S.V. On similarity of 2×2 matrices over the ring of Gaussian integers with reducible characteristic polynomial// Vestn. Nizhni Novgorod Univ. – 2008. – V. 4. – P. 122–126. (in Russian)

Velichko I.N. Generalized Kloosterman sum over the matrix ring Mn(Z[i])// Visn. Odes. Nats. Univ., Ser. Mat. and Mekh. – 2010. – V.1, №19. – P. 9–20. (in Russian)

Savastru O., Varbanets S. Norm Kloosterman sums over Z[i]// Algebra Discrete Math. – 1995. – V.80. – P. 105–137.

Taylor G. Cyclotomic matrices and graphs over the ring of integers of some imaginary quadratic fields// J. Algebra. – 2011. – V.331. – P. 523–545.

Greaves G. Cyclotomic matrices over the Eisenstein and Gaussian integers// J. Algebra. – 2012. – V.372. – P. 560–583.

Ladzoryshyn N.B. On equivalence of pairs of matrices, in which determinants are orders of primes, over quadratic Euclidean rings// Carpathian Math. Publ. – 2013. – V.5, №1. – P. 63–69, https://doi:10.15330/cmp.5.1.63-69. (in Ukrainian)

Ladzoryshyn N., Petrychkovych V. Equivalence of pairs of matrices with relatively prime determinants over quadratic rings of principal ideals// Bul. Acad. ¸Stiin¸te Repub. Mold. Mat. – 2014. – №3. – P. 38–48.

Ladzoryshyn N.B., Petrychkovych V.M. Standard form of matrices over quadratic rings with respect to the (z, k)–equivalence and the structure of solutions of bilateral matrix linear equations// J. Math. Sci. – 2021. – V.253., №1. – P. 54–62. https://doi.org/10.1007/s10958-021-05212-w

Ladzoryshyn N.B., Petrychkovych V.M., Zelisko H.V. Matrix Diophantine equations over quadratic rings and their solutions // Carpathian Math. Publ. – 2020. – V.12, №2. – P. 368–375. https://doi.org/10.15330/cmp.12.2.368-375

Petrychkovych V.M., Zelisko H.V., Ladzoryshyn N.B. The standard form of matrices over the ring of Gaussian integers with respect to (z, k)–equivalence // Appl. Probl. Mech. and Math. – 2020. – V. 18. – P. 5–10. https://doi.org/10.15407/apmm2020.18.5-10 (in Ukrainian)