Email this article On the Similarity over the Ring of Integers of Certain Nilpotent Matrices of Maximal Rank
- Authors: Sidorov S.V.1, Utkin G.V.1
-
Affiliations:
- National Research Lobachevsky State University of Nizhny Novgorod
- Issue: Vol 25, No 4 (2023)
- Pages: 284-298
- Section: Mathematics
- Published: 24.11.2023
- URL: https://journals.rcsi.science/2079-6900/article/view/360836
- DOI: https://doi.org/10.15507/2079-6900.25.202304.284-298
- ID: 360836
Cite item
Full Text
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 superdiag(a1,a2,...,an-1) with a single non-zero superdiagonal, that is a generalization of the Jordan cell Jn(0)=superdiag(1,1,...,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 a1,a2,...,an-1 of the first superdiagonal of matrix A, it is proven that the matrix A is similar to matrix superdiag(a1,a2,...,an-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 superdiag(a1,a2,...,an-1).
About the authors
Sergey V. Sidorov
National Research Lobachevsky State University of Nizhny Novgorod
Email: sesidorov@yandex.ru
ORCID iD: 0000-0003-2883-6427
German V. Utkin
National Research Lobachevsky State University of Nizhny Novgorod
Author for correspondence.
Email: german.utkingu@gmail.com
ORCID iD: 0000-0003-4794-2591
References
- R.A. Sarkisjan, "Conjugacy problem for sets of integral matrices", Math. Notes, 25:6 (1979), 419–426. DOI: https://doi.org/10.1007/BF01230982
- F. J.Grunewald, Solution of the conjugacy problem in certain arithmetic groups, Word Problems II, 95, 1980, 101–139 DOI: https://doi.org/10.1016/S0049-237X(08)71335-1.
- B.Eick, T.Hofmann, E.A.O’Brien, "The conjugacy problem in GL(n, Z)", J. Lond. Math. Soc., 100:3 (2019), 731–756. DOI: https://doi.org/10.1112/jlms.12246
- D.Husert, Similarity of integer matrices, PhD Thesis, University of Paderborn, 2017, 147 p.
- S.Marseglia, "Computing the ideal class monoid of an order", J. Lond. Math. Soc., 101:3 (2019), 984–1007. DOI: https://doi.org/10.1112/jlms.12294
- J.Opgenorth, W.Plesken, T. Schulz, "Crystallographic algorithms and tables", Acta Cryst. Sect., 54:5 (1998), 517–531. DOI: https://doi.org/10.1107/S010876739701547X
- O.Karpenkov, "Multidimensional Gauss reduction theory for conjugacy classes of SL(n, Z)", J. Theor. Nombres Bordeaux, 25:1 (2013), 99–109.
- S.V. Sidorov, "Similarity of matrices with integer spectra over the ring of integers", Russian Math. (Iz. VUZ), 55:3 (2011), 77–84 (In Russ.).
- H.Appelgate, H.Onishi, "The similarity problem 3×3 integer matrices", Linear Algebra Appl., 42:2 (1982), 159–174. DOI: https://doi.org/10.2307/2043695
- S.V. Sidorov, "On similarity of matrices of third order over the ring of integers with reducible characteristic polynomial", Vestnik Nizhegorodsk. Univ., 2009, no. 1, 119–127 (In Russ.).
- S.V. Sidorov, Selection of effectively solvable classes in the problem of similarity of matrices over the ring of integers, PhD Dissertation, Nizhny Novgorod, 2015 (In Russ.).
- V.N. Shevchenko, S.V. Sidorov, "On the similarity of second-order matrices over the ring of integers", Russian Math. (Iz. VUZ), 50:4 (2006), 56–63 (In Russ.).
- S.V. Sidorov, E.E.Chilina, "On non-hyperbolic algebraic automorphisms of the torus", Zhurnal SVMO, 23:3 (2021), 295–307 (In Russ.). DOI: https://doi.org/10.15507/2079.6900.23.202103.295-307
- V.V.Gorbatsevich, "Compact solvmanifolds of dimension at most ≤ 4", Sib. Math. J, 50:2 (2009), 239–252.
- S.V. Sidorov, "On the similarity of certain integer matrices with single eigenvalue over the ring of integers", Math Notes, 105 (2019), 756–762 (In Russ.). DOI: https://doi.org/10.1134/S0001434619050122
- M.Newman, Integral matrices, Academic Press, New York, 1972, 223 p.
- F. Lazebnik, "On systems of linear Diophantine equations", Mathematics Magazine, 69 (1996), 261–266. DOI: https://doi.org/10.2307/2690528
- A. Schrijver, Theory of linear and integer programming, Wiley, 1998, 464 p.
Supplementary files
