DOI 10.15507/2079-6900.25.202304.284-298

*Original article*

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

*MSC2020 15A04, 15A18, 15A21, 15B36*

#### On the Similarity over the Ring of Integers of Certain Nilpotent Matrices of Maximal Rank

**S. V. Sidorov, G. V. Utkin**

*National Research Lobachevsky State University of Nizhny Novgorod (Nizhny Novgorod, Russian Federation)*

**Abstract.** This paper is devoted to the problem of matrix similarity recognition over the ring of integers for some families of matrices. Namely, nilpotent upper triangular matrices of maximal rank are considered such that only first and second superdiagonals of these matrices are non-zero. Several necessary conditions are obtained for similarity of such matrices to matrices of the form $\mathrm{superdiag}(a_1,a_2,\ldots,a_{n-1})$ with a single non-zero superdiagonal, that is a generalization of the Jordan cell $J_n(0)=\mathrm{superdiag}(1,1,\ldots,1)$. These conditions are formulated in simple terms of divisibility and greatest common divisors of matrix elements. The result is obtained by reducing the problem of similarity recognition to the problem of solving in integers a system of linear equations and applying the known necessary similarity conditions for arbitrary matrices. Under some additional conditions on the elements $a_1,a_2,\ldots,a_{n-1}$ of the first superdiagonal of matrix $A$, it is proven that the matrix $A$ is similar to matrix $\mathrm{superdiag}(a_1,a_2,\ldots,a_{n-1})$ regardless of the values of the elements of the second superdiagonal. Moreover, for the considered matrices of the third and the fourth orders, easily verifiable necessary and sufficient similarity conditions are obtained describing their similarity to a matrix of the form $\mathrm{superdiag}(a_1,a_2,\ldots,a_{n-1})$.

**Key Words:** similarity of matrices, Jordan form, Smith normal diagonal form, ring of integers, nilpotent matrix

**For citation:** *S. V. Sidorov, G. V. Utkin.* On the Similarity over the Ring of Integers of Certain Nilpotent Matrices of Maximal Rank. *Zhurnal Srednevolzhskogo matematicheskogo obshchestva*. 25:4(2023), 284–298. DOI: https://doi.org/10.15507/2079-6900.25.202304.284-298

*
Submitted: 04.08.2023;
Revised: 05.10.2023;
Accepted: 24.11.2023 *

*Information about the authors:*

**Sergey V. Sidorov**, Associate Professor, Department of Algebra, Geometry and Discrete Mathematics, National Research Lobachevsky State University of Nizhny Novgorod (23 Gagarina Av., Nizhny Novgorod 603022, Russia), Ph.D. (Phys.-Math.), ORCID: https://orcid.org/0000-0003-2883-6427, sesidorov@yandex.ru

**German V. Utkin**, Laboratory Assistant, Department of Algebra, Geometry and Discrete Mathematics, National Research Lobachevsky State University of Nizhny Novgorod (23 Gagarina Av., Nizhny Novgorod 603022, Russia), Bachelor of Applied Computer Science, ORCID: https://orcid.org/0000-0003-4794-2591, german.utkingu@gmail.com

*All authors have read and approved the final manuscript.*

*Conflict of interest: The authors declare no conflict of interest.*