Sidorov, S. V. Similarity of matrices with integer spectrum over the ring of integers. (English. Russian original) Zbl 1232.15016 Russ. Math. 55, No. 3, 77-84 (2011); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2011, No. 3, 86-94 (2011). The notion of similarity of matrices over the ring of integers \(\mathbb Z\) is a natural generalization of the notion of similarity of matrices over the field of rational numbers \(\mathbb Q\). A matrix \(B \in {\mathbb Z}^{n \times n}\) over \(\mathbb Z\) is said to be similar to a matrix \(A \in {\mathbb Z}^{n \times n}\) if there exists a unimodular matrix (i. e., a matrix with the determinant equal to 1 or -1) \(S \in {\mathbb Z}^{n \times n}\) such that \(B= S^{-1} A S\). The problem is to find whether two integer matrices are similar over \(\mathbb Z\). In this paper, a quasipolynomial-time algorithm (i. e., polynomial-time for a fixed dimension) is given for recognizing similarity of matrices over \(\mathbb Z\) for the class of matrices with integer spectrum whose Jordan forms contain no blocks of the same order for one and the same eigenvalues. Then, it is proved for matrices all of whose eigenvalues are different that the number of similarity classes is finite and an estimate of the number of similarity classes is given. Reviewer: Václav Burjan (Praha) Cited in 5 Documents MSC: 15A21 Canonical forms, reductions, classification 15B36 Matrices of integers 65F30 Other matrix algorithms (MSC2010) Keywords:similarity of matrices; ring of integers; Jordan canonical form; matrix spectrum; quasipolynomial-time algorithm; eigenvalues PDFBibTeX XMLCite \textit{S. V. Sidorov}, Russ. Math. 55, No. 3, 77--84 (2011; Zbl 1232.15016); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2011, No. 3, 86--94 (2011) Full Text: DOI References: [1] A. I. Maltsev, Foundations of Linear Algebra (Nauka, Moscow, 1970) [in Russian]. [2] V. N. Shevchenko and S. V. Sidorov, ”On Similarity of Second Order Matrices Over the Ring of Whole Numbers,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 4, 57–64 (2006) [Russian Mathematics (Iz. VUZ) 50 (4), 56–63 (2006)]. [3] R. A. Sarkisyan, ”Conjugacy Problem for Collections of IntegerMatrices,” Matem. Zametki 25(6), 811–824 (1979). · Zbl 0407.10046 [4] F. Grunewald, ”Solution of the Conjugacy Problem in Certain Arithmetic Groups,” Word problems II, Stud. Logic Found. Math. 95, 101–139 (1980). [5] F. R. Gantmakher The Theory of Matrices (Fizmatlit, Moscow, 2004) [in Russian]. 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.