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

Abstract

Given two n× n integral matrices A and B, they are said to be equivalent if B= S −1 AS, where S is an n× n integral matrix with determinant ±1. If we consider n× n integral matrices with a fixed characteristic polynomial that is irreducible over Q , it is well known from a result by Latimer and MacDuffee that the number of matrix classes (equivalence classes of matrices) is equal to the number of ideal classes ( I ≅ J if I= qJ for some q in the quotient field) of the ring obtained by adjoining a root of the characteristic polynomial to Z . In this paper, we develop an effective version of this result for 2×2 matrices. We present an algorithm which given a 2×2 matrix finds a canonical representative in its class. In particular this allows us to determine whether two matrices are equivalent.