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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.

MSC:

15A21 Canonical forms, reductions, classification
15B36 Matrices of integers
65F30 Other matrix algorithms (MSC2010)
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References:

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