##plugins.themes.bootstrap3.article.sidebar##
##plugins.themes.bootstrap3.article.main##
Abstract
The structure of matrices over a domain of principal ideals with respect to similarity transformations is investigated. The article consists of four chapters. In the second section supporting results are presented.
In particular, we construct the triangular form of the matrix with respect to the similarity transformation for the case when its minimal polynomial decomposes into the product of different linear factors. In Section 3 it is proved that the Hessenberg form of a matrix A with an irreducible minimal quadratic polynomial m(λ) is a block triangular matrix with blocks of dimensional 2х2 with characteristic polynomials m(λ) on the main diagonal. In the fourth section it is proved that a matrix A with the minimal polynomial m (λ) = (λ-α) (λ-β), α ≠ β, is similar to the lower block-triangular matrix, the diagonal blocks of which are diagonal matrices with elements α and β on the main diagonals respectively. As a result of this the canonical form of involutory matrices over the ring of integers with respect to similarity transformations is established.
##plugins.themes.bootstrap3.article.details##
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution (CC-BY) 4.0 License that allows others to share the work with an acknowledgment of the work’s authorship and initial publication in this journal.
Provided they are the owners of the copyright to their work, authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal’s published version of the work (e.g., post it to an institutional repository, in a journal or publish it in a book), with an acknowledgment of its initial publication in this journal.
Authors are permitted and encouraged to post their work online (e.g., in institutional repositories, disciplinary repositories, or on their website) prior to and during the submission process.
References
2. A. Behn, A. B. Van der Merwe. An algorithmic version of the theorem by \\rm Latimer and \\rmMacDuffee for 2\\times2 integral matrices. Linear Algebra Appl., 346(1-3):1-14, 2002.,
3. W. С. Brown. Matrices over commutative rings, volume 169 of Pure and Applied Mathematics. New York: Marcel Dekker, 1993.,
4. J. T. Campbell, E. C. Trouy. When are two elements of \\rm GL(2, Z) similar? Linear Algebra Appl., 157:175-184, 1991.,
5. J. D. Dixon. An isomorphism criterion for modules over a principal ideal domain. Linear Multilin. Algebra, 8(1):69-72, 1979.,
6. Sh. Friedland. Matrices: Algebra, Analysis and Applications. World Scientific, 2015.,
7. R. M. Guralnick. Similarity of matrices over commutative rings. Linear Algebra Appl., 157:55-68, 1991.,
8. W. H. Gustafson. Roth's theorems over commutative rings. Linear Algebra Appl., 23:245-251, 1979.,
9. W. H. Gustafson. On matrix similarity over commutative rings. Linear Multilin. Algebra, 10(3):249-252, 1981.,
10. C. G. Latimer, C. C. MacDuffee. A correspondence between classes of ideals and classes of matrices. Annals of Math. Second Series., 34(2):313-316, 1933.,
11. B. R. McDonald. Similarity of matrices over artinian principal ideal rings. Linear Algebra Appl., 21(2):153-162, 1978.,
12. M. Newman. Integer Matrices. New York: Acad.Press, 1972.,
13. O. Taussky. Matrices of rational integers. Bull. Amer. Math. Soc., 66(5):327-345, 1960.,
14. H. K. Wimmer. Problem 45-7, \\rm A generalization of \\rmRoth’s theorem. Bull. Internat. Linear Algebra Society, 45:48, 2010.,
15. В. М. Бондаренко. О подобии матриц над кольцами классов вычетов. Математический сборник. \\rmК.: \\rmНаукова думка, 1:275-277, 1976.,
16. Ф. Р. Гантмахер. Теория матриц. М.: Наука, 1978.,
17. Ф. Груневальд, Н. К. Иыуду. Задача сопряжения для 2\\times 2-матриц над кольцами многочленов. Современная математика и ее прилож., 30(3):31-45, 2005.,
18. П. М. Гудивок. О модулярных и целочисельных представлениях конечных групп. ДАН СССР, 214(5):993-996, 1974.,
19. Ю. А. Дрозд. О ручных и диких матричных задачах. Матричные задачи, 1:104-114, 1977.,
20. Ю. А. Дрозд. Ручные и дикие матричные задачи. Представления и квадратичные формы, 1:39-74, 1979.,
21. А. А. Нечаев. О подобии матриц над коммутативным локальным артиновым кольцом. Труды сем. им. И. Г. Петровского, 9:81-101, 1989.,
22. В. М. Прокіп. Діагоналізованість матриць над областю головних ідеалів з мінімальним многочленом m(\\lambda)=(\\lambda -\\alpha)(\\lambda -\\beta), (\\alpha \\ne \\beta). Укр. мат. вісник, 7(2):212-219, 2010.,
23. В. М. Прокіп. Діагоналізовність матриць над областю головних ідеалів. Укр. мат. журнал, 64(2):283-288, 2012.,
24. В. М. Прокіп. Структура матриць рангу один над областю головних ідеалів відносно перетворення подібності. Мат. методи та фіз.-мех. поля, 59(3):68-76, 2017.,
25. С. В. Сидоров, В. Н. Шевченко. О подобии матриц второго порядка над кольцом целых чисел. Изв. вузов. Математика, 4:57-64, 2006.,
26. Д. A. Супруненко. О сопряженности матриц над кольцом вычетов. ДАН БССР, 8(11):693-695, 1964.