An algorithmic version of the theorem by Latimer and MacDuffee for 2×2 integral matrices

Behn, Antonio; Merwe, A.B. Van der

doi:10.1016/s0024-3795(01)00518-3

Cited by 7 publications

(5 citation statements)

References 3 publications

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“…All matrices of type (2.1) are (not) uniquely clean iff the reduced representative is so. This is (see [4], p. 7) −1 −1 1 0 and it is readily seen that this matrix is not uniquely clean. It has 3 nontrivial clean decompositions:…”

Section: The Elliptic Casementioning

confidence: 99%

“…The proof follows the same lines as the proof of Theorem 2.1. The characteristic polynomial for such matrices is (X − 1) 2 , so factors over Z and it suffices to deal with the reduced representative, which is now V = −1 1 0 −1 (we just use the algorithm described by [4], in the proof of Theorem 1.5, for example for 0…”

Section: The Parabolic Casementioning

confidence: 99%

“…However, it was not necessary to use such results because of the transfer done directly to similarity classes of integral 2 × 2 matrices done in Behn, Van der Merwe paper (see [4]). From this paper we recall the following definitions and results.…”

Section: Introductionmentioning

confidence: 99%

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Uniquely clean 2×2 invertible integral matrices

Andrica¹,

Călugăreanu²

2017

Stud. Univ. Babes-Bolyai Math.

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Abstract. While units in any unital ring are strongly clean by definition, which units are uniquely clean, is a far from being simple question, even in particular rings. In this paper, the question is solved for 2 × 2 integral matrices. It turns out that uniquely clean invertible matrices are scarce: only the matrices similar to 1 0 0 −1 . The study is splitted into three cases: the elliptic, the parabolic and the hyperbolic cases, according to the discriminant of their characteristic polynomial. In the first two cases, units are not uniquely clean.Mathematics Subject Classification (2010): 15B36, 16U99, 11D09, 11D45.

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Section: The Elliptic Casementioning

confidence: 99%

Section: The Parabolic Casementioning

confidence: 99%

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Uniquely clean 2×2 invertible integral matrices

Andrica¹,

Călugăreanu²

2017

Stud. Univ. Babes-Bolyai Math.

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“…Using this observation, Sarkisyan [Sar79] and Grunewald [Gru80] independently showed that the conjugacy problem over for arbitrary pairs of matrices is decidable. Moreover, Applegate and Onishi [AO81, AO82] considered the cases of and matrices, and Behn and Van der Merwe [BVdM02] also considered the case.…”

Section: Application: Similarity Of Matrices Over Rings Of Integersmentioning

confidence: 99%

Computation of lattice isomorphisms and the integral matrix similarity problem

Bley

Hofmann

Johnston

2022

Forum of Mathematics, Sigma

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Let K be a number field, let A be a finite-dimensional K-algebra, let $\operatorname {\mathrm {J}}(A)$ denote the Jacobson radical of A and let $\Lambda $ be an $\mathcal {O}_{K}$ -order in A. Suppose that each simple component of the semisimple K-algebra $A/{\operatorname {\mathrm {J}}(A)}$ is isomorphic to a matrix ring over a field. Under this hypothesis on A, we give an algorithm that, given two $\Lambda $ -lattices X and Y, determines whether X and Y are isomorphic and, if so, computes an explicit isomorphism $X \rightarrow Y$ . This algorithm reduces the problem to standard problems in computational algebra and algorithmic algebraic number theory in polynomial time. As an application, we give an algorithm for the following long-standing problem: Given a number field K, a positive integer n and two matrices $A,B \in \mathrm {Mat}_{n}(\mathcal {O}_{K})$ , determine whether A and B are similar over $\mathcal {O}_{K}$ , and if so, return a matrix $C \in \mathrm {GL}_{n}(\mathcal {O}_{K})$ such that $B= CAC^{-1}$ . We give explicit examples that show that the implementation of the latter algorithm for $\mathcal {O}_{K}=\mathbb {Z}$ vastly outperforms implementations of all previous algorithms, as predicted by our complexity analysis.

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“…. As is known, the conjugacy classes in SL(2, Z) are not determined only by the trace and there are finitely many conjugacy classes with the same trace (see [1] and its references).…”

Section: Unitary Representations For Closed 3-manifoldsmentioning

confidence: 99%