Avni, Nir; Onn, Uri; Prasad, Amritanshu; Vaserstein, Leonid Similarity classes of \(3 \times 3\) matrices over a local principal ideal ring. (English) Zbl 1176.15013 Commun. Algebra 37, No. 8, 2601-2615 (2009). In general, the classification problem of similarity classes in \(n\) by \(n\) matrices over rings other than fields is a highly nontrivial question. In this article, similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly. Reviewer: Sheng Chen (Harbin) Cited in 9 Documents MSC: 15A21 Canonical forms, reductions, classification 15A54 Matrices over function rings in one or more variables 15B33 Matrices over special rings (quaternions, finite fields, etc.) Keywords:enumeration; classification; similarity classes; local principal ideal commutative ring; generating function PDFBibTeX XMLCite \textit{N. Avni} et al., Commun. Algebra 37, No. 8, 2601--2615 (2009; Zbl 1176.15013) Full Text: DOI arXiv References: [1] DOI: 10.1016/0024-3795(82)90146-X · Zbl 0484.15011 [2] Appelgate H., Proc. Amer. Math. Soc. 87 pp 233– (1983) [3] Avni N., Representation Zeta Functions for SL 3 (2007) [4] DOI: 10.1215/S0012-7094-68-03506-0 · Zbl 0157.10402 [5] Dickson L. E., Algebraic Theories (1959) · Zbl 0086.01103 [6] DOI: 10.2307/1970321 · Zbl 0115.39004 [7] DOI: 10.2307/1970342 · Zbl 0126.16704 [8] Grunewald , F. J. ( 1980 ).Solution of the Conjugacy Problem in Certain Arithmetic Groups. Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), Stud. Logic Foundations Math. , Vol. 95 , pp. 101 – 139 , North-Holland : Amsterdam . [9] Hoffman K., Linear Algebra., 2. ed. (1971) [10] Howe R. E., Pacific J. Math. 73 pp 365– (1977) [11] Kurakin V. L., Mat. Zametki 80 pp 403– (2006) [12] Nagornyĭ S. V., Zap. Naučn. Sem. Leningrad. Ordel. Mat. Inst. Steklov. (LOMI). 75 pp 143– (1978) [13] Nechaev A. A., Trudy Sem. Petrovsk. 9 pp 81– (1983) [14] DOI: 10.1007/BF01351597 [15] Onn U., Representations of Automorphism Groups of Rank Two Finite -Modules (2007) [16] DOI: 10.1016/0024-3795(83)90204-5 · Zbl 0516.15005 [17] DOI: 10.1090/S0002-9939-1973-0309963-X This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.