On the Similarity of Certain Integer Matrices with Single Eigenvalue over the Ring of Integers

S. V. Sidorov

Matematičeskie zametki, Tome 105 (2019) no. 5, pp. 763-770.

Résumé

The problem of the similarity of integer matrices with single eigenvalue over the ring of integers is considered. A criterion for a matrix to be similar to a Jordan block is obtained. In addition, a similarity criterion for matrices whose minimal polynomial has degree 2 is obtained.

Détail

Keywords: similarity of matrices, Smith normal diagonal form, ring of integers.
Mots-clés : Jordan form

Bibliographie

[1] M. Newman, Integral Matrices, Academic Press, New York, 1972 | MR | Zbl

[2] A. I. Maltsev, Osnovy lineinoi algebry, Nauka, M., 1970 | MR

[3] D. A. Suprunenko, “O sopryazhennosti matrits nad koltsom vychetov”, Dokl. AN BSSR, 8 (1964), 693–695 | MR | Zbl

[4] R. W. Davis, “Certain matrix equations over rings of integers”, Duke Math. J., 35 (1968), 49–59 | DOI | MR | Zbl

[5] J. Pomfret, “Similarity of matrices over finite rings”, Proc. Amer. Math. Soc., 37:2 (1973), 421–422 | DOI | MR | Zbl

[6] W. H. Gustafson, “On matrix similarity over commutative rings”, Linear and Multilinear Algebra, 10:3 (1981), 249–252 | DOI | MR | Zbl

[7] A. A. Nechaev, “O podobii matrits nad kommutativnym lokalnym artinovym koltsom”, Tr. sem. im. I. G. Petrovskogo, 9 (1983), 81–101 | MR | Zbl

[8] R. M. Guralnick, “Similarity of matrices over commutative rings”, Linear Algebra Appl., 157 (1991), 55–68 | DOI | MR | Zbl

[9] V. Prokip, “On similarity of matrices over commutative rings”, Linear Algebra Appl., 399 (2005), 225–233 | DOI | MR | Zbl

[10] V. L. Kurakin, “Invarianty podobiya matrits nad kommutativnym artinovym tsepnym koltsom”, Matem. zametki, 80:3 (2006), 403–412 | DOI | MR | Zbl

[11] V. M. Bondarenko, M. Yu. Bortos, R. F. Dinis, A. A. Tylyshchak, “Reducibility and irreducibility of monomial matrices over commutative rings”, Algebra Discrete Math., 16:2 (2013), 171–187 | MR | Zbl

[12] N. A. Vavilov, A. V. Stepanov, “Lineinye gruppy nad obschimi koltsami I. Obschie mesta”, Voprosy teorii predstavlenii algebr i grupp. 22, Zap. nauchn. sem. POMI, 394, POMI, SPb., 2011, 33–139 | MR

[13] P. M. Gudivok, “O modulyarnykh i tselochislennykh predstavleniyakh konechnykh grupp”, Dokl. AN SSSR, 214:5 (1974), 993–996 | MR | Zbl

[14] V. M. Bondarenko, “O podobii matrits nad koltsom klassov vychetov”, Matematicheskii sbornik, Naukova dumka, Kiev, 1976, 275–277

[15] S. V. Nagornyi, “Kompleksnye predstavleniya polnoi lineinoi gruppy tretei stepeni nad koltsom vychetov po primarnomu modulyu”, Koltsa i lineinye gruppy, Zap. nauchn. sem. LOMI, 75, Izd-vo «Nauka», Leningrad. otd., L., 1978, 143–150 | MR | Zbl

[16] Yu. A. Drozd, “O ruchnykh i dikikh matrichnykh zadachakh”, Matrichnye zadachi, In-t matem. AN USSR, Kiev, 1977, 104–114 | Zbl

[17] V. M. Bondarenko, M. Yu. Bortosh, “Indecomposable and isomorphic objects in the category of monomial matrices over a local ring”, Ukrainian Math. J., 69:7 (2017), 1034–1050 | DOI | MR

[18] V. V. Gorbatsevich, “O kompaktnykh solvmnogoobraziyakh razmernosti $\le4$”, Sib. matem. zhurn., 50:2 (2009), 300–319 | MR | Zbl

[19] V. N. Shevchenko, S. V. Sidorov, “O podobii matrits vtorogo poryadka nad koltsom tselykh chisel”, Izv. vuzov. Matem., 2006, no. 4, 57–64 | MR | Zbl

[20] S. V. Sidorov, “O podobii matrits s tselochislennym spektrom nad koltsom tselykh chisel”, Izv. vuzov. Matem., 2011, no. 3, 86–94 | MR | Zbl

[21] S. V. Sidorov, Vydelenie effektivno razreshimykh klassov v zadache podobiya matrits nad koltsom tselykh chisel, Dis. ... kand. fiz.-matem. nauk, Nizhnii Novgorod, 2015

[22] S. V. Sidorov, “O podobii matrits tretego poryadka nad koltsom tselykh chisel, imeyuschikh privodimyi kharakteristicheskii mnogochlen”, Vestn. Nizhegorodskogo un-ta im. N. I. Lobachevskogo, 2009, no. 1, 119–127