Надіслати цю статтю Про стандартні форми пар матриць над кільцем цілих ґауссових чисел відносно (z,k)–еквівалентності
Анотація
Досліджується (z,k)-еквівалентність пар матриць над кільцем цілих ґауссових чисел і їх звідність до стандартних форм. Встановлено, що число стандартних форм пар матриць над цим кільцем є скінченним. Наведено класи пар матриць з мінімальними і максимальними числами стандартних форм.
Зразок для цитування: Н. Б. Ладзоришин, В. М. Петричкович, “Про стандартні форми пар матриць над кільцем цілих ґауссових чисел відносно (z,k)-еквівалентності,” Мат. методи та фіз.-мех. поля, 64, No. 4, 18–24 (2021), https://doi.org/10.15407/mmpmf2021.64.4.18-24
Translation: N. B. Ladzoryshyn, V. M. Petrychkovych, “On standard forms of the pairs of matrices over the ring of Gaussian integers with respect to the (z,k)-equivalence,” J. Math. Sci., 279, No. 2, 133–140 (2024), https://doi.org/10.1007/s10958-024-07000-8
Ключові слова
Посилання
I. N. Velichko, “Generalized Klosterman sums over the matrix ring Mn(Z[i]),” Visn. Odes’k. Nats. Univ. Mat. Mekh., 15, No. 19, 9–20 (2010) (in Russian).
V. R. Zelisko, N. B. Ladzoryshyn, V. M. Petrychkovych, “On the equivalence of matrices over Euclidean quadratic rings,” Prykl. Probl. Mekh. Mat., Issue 4, 16–21 (2006) (in Ukrainian).
P. S. Kazimirskii, Decomposition of Matrix Polynomials into Factors [in Ukrainian], Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv (2015) (in Ukrainian).
N. B. Ladzoryshyn, “On the equivalence of pairs of matrices whose determinants are powers of prime numbers over Euclidean quadratic rings,” Karpat. Mat. Publ., 5, No. 1, 63−69 (2013) (in Ukrainian), https://doi.org/10.15330/cmp.5.1.63-69
N. B. Ladzoryshyn, V. M. Petrychkovych, “Standard form of matrices over quadratic rings with respect to the (z,k)-equivalence and the structure of solutions of bilateral matrix linear equations,” Mat. Met. Fiz.-Mekh. Polya, 61, No. 2, 49–56 (2018); English translation: J. Math. Sci., 253, No. 1, 54–62 (2021), https://doi.org/10.1007/s10958-021-05212-w
V. M. Petrichkovich, “Semiscalar equivalence and the Smith normal form of polynomial matrices,” Mat. Met. Fiz.-Mekh. Polya, Issue 26, 13–16 (1987); English translation: J. Sov. Math., 66, No. 1, 2030–2033 (1993), https://doi.org/10.1007/BF01097386
V. M. Petrychkovych, Generalized Equivalence of Matrices and Their Collections and Factorization of Matrices over Rings [in Ukrainian], Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv (2015) (in Ukrainian).
V. M. Petrychkovych, H. V. Zelisko, N. B. Ladzoryshyn, “Standard form of matrices over the ring of Gaussian integers with respect to the (z,k)-equivalence,” Prykl. Probl. Mekh. Mat., Issue 18, 5–10 (2020) (in Ukrainian), https://doi.org/10.15407/apmm2020.18.5-10
L. Cossu and P. Zanardo, “Idempotent factorizations of singular 2x2-matrices over quadratic integer rings,” Linear Multilinear Algebra, 70, No. 2, 297–309 (2022), https://doi.org/10.1080/03081087.2020.1721416
N. S. Dzhaliuk, V. M. Petrychkovych, “Solutions of the matrix linear bilateral polynomial equation and their structure,” Algebra Discrete Math., 27, No. 2, 243–251 (2019).
N. S. Dzhaliuk, V. M. Petrychkovych, “The matrix linear unilateral and bilateral equations with two variables over commutative rings,” Int. Sch. Res. Notices. ISRN Algebra, Article ID 205478 (2012), https://doi.org/10.5402/2012/205478
N. Ladzoryshyn, V. Petrychkovych, “Equivalence of pairs of matrices with relatively prime determinants over quadratic rings of principal ideals,” Bul. Acad. Ştiinţe Repub. Mold. Mat., 76, No. 3, 38–48 (2014).
N. B. Ladzoryshyn, V. M. Petrychkovych, H. V. Zelisko, “Matrix Diophantine equations over quadratic rings and their solutions,” Karpat. Mat. Publ., 12, No. 2, 368–375 (2020), https://doi.org/10.15330/cmp.12.2.368-375
B. Nica, “The unreasonable slightness of $E_2$ over imaginary quadratic rings,” Am. Math. Month., 118, No. 5, 455–462 (2011), https://doi.org/10.4169/amer.math.monthly.118.05.455
V. Petrychkovych, “Generalized equivalence of pairs of matrices,” Linear Multilinear Algebra, 48, No. 2, 179–188 (2000), https://doi.org/10.1080/03081080008818667
V. Petrychkovych, “Standard form of pairs of matrices with respect to generalized equivalence,” Visn. L’viv. Univ. Ser. Mekh.-Mat., Issue 61, 148–155 (2003).
S. P. Varbanets, “General Kloosterman sums over the ring of Gaussian integers,” Ukr. Mat. Zh., 59, No. 9, 1179–1200 (2007); Reprinted in: Ukr. Math. J., 59, No. 9, 1313–1341 (2007), https://doi.org/10.1007/s11253-007-0090-4
I. N. Velichko, “Generalized Kloosterman sum over the ring Mn(Z),” Siauliai Math. Semin., 5, No. 13, 121–136 (2010).
Посилання
- Поки немає зовнішніх посилань.
Ця робота ліцензована Creative Commons Attribution 3.0 License.